Abstract

We derive a pulse propagation equation for a graphene-clad optical fiber, treating the optical response of the graphene and nonlinearity of the dielectric fiber core as perturbations in asymptotic expansion of Maxwell equations. We analyze the effective nonlinear and attenuation coefficients due to the graphene layer. Based on the recent experimental measurements of the nonlinear graphene conductivity, we predict considerable enhancement of the effective nonlinearity for subwavelength fiber core diameters.

© 2013 Optical Society of America

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2013

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

S. Hong, J. I. Dadap, N. Petrone, P. Yeh, J. Hone, and R. M. Osgood, Phys. Rev. X 3, 021014 (2013).
[CrossRef]

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

H. Dong, C. Conti, A. Marini, and F. Biancalana, J. Phys. B 6, 155401 (2013).
[CrossRef]

A. V. Gorbach, Phys. Rev. A 87, 013830 (2013).
[CrossRef]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

X. Zhao, A. V. Gorbach, and D. V. Skryabin, J. Opt. Soc. Am. B 30, 812 (2013).
[CrossRef]

2012

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

S. Chu, S. Wang, and Q. Gong, Chem. Phys. Lett. 523, 104 (2012).
[CrossRef]

H. Zhang, S. Virally, Q. Bao, L. Kian Ping, S. Massar, N. Godbout, and P. Kockaert, Opt. Lett. 37, 1856 (2012).
[CrossRef]

Q. Bao and K. P. Loh, ACS Nano 6, 3677 (2012).
[CrossRef]

2011

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

2010

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

J. J. Dean and H. V. van Driel, Phys. Rev. B 82, 125411 (2010).
[CrossRef]

2009

2008

S. A. Mikhailov and K. Ziegler, J. Phys. 20, 384204 (2008).
[CrossRef]

2007

S. A. Mikhailov, Europhys. Lett. 79, 27002 (2007).
[CrossRef]

2003

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

2000

Afshar, S. V.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Anderson, T.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Bao, Q.

Bezares, F. J.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Biancalana, F.

H. Dong, C. Conti, A. Marini, and F. Biancalana, J. Phys. B 6, 155401 (2013).
[CrossRef]

Birks, T. A.

Bonaccorso, F.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Brauer, A.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Bravo-Abad, J.

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

Bulovic, V.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Caldwell, J. D.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Chen, X.

Chiu, H.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Chu, S.

S. Chu, S. Wang, and Q. Gong, Chem. Phys. Lett. 523, 104 (2012).
[CrossRef]

Chun, H.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Conti, C.

H. Dong, C. Conti, A. Marini, and F. Biancalana, J. Phys. B 6, 155401 (2013).
[CrossRef]

Currie, M.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Dadap, J. I.

Dean, J. J.

J. J. Dean and H. V. van Driel, Phys. Rev. B 82, 125411 (2010).
[CrossRef]

Dong, H.

H. Dong, C. Conti, A. Marini, and F. Biancalana, J. Phys. B 6, 155401 (2013).
[CrossRef]

Dresselhaus, M.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Dulkeith, E.

Ferrari, A. C.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Garcia-Vidal, F. J.

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

Gerstenkorn, C.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Godbout, N.

Gong, Q.

S. Chu, S. Wang, and Q. Gong, Chem. Phys. Lett. 523, 104 (2012).
[CrossRef]

Gorbach, A. V.

A. V. Gorbach, Phys. Rev. A 87, 013830 (2013).
[CrossRef]

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

X. Zhao, A. V. Gorbach, and D. V. Skryabin, J. Opt. Soc. Am. B 30, 812 (2013).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Green, W. M.

Gu, T.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Hale, P. J.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

Hartley, R.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Hasan, T.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Hendry, E.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

Ho, J. Z.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Hone, J.

S. Hong, J. I. Dadap, N. Petrone, P. Yeh, J. Hone, and R. M. Osgood, Phys. Rev. X 3, 021014 (2013).
[CrossRef]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Hong, S.

S. Hong, J. I. Dadap, N. Petrone, P. Yeh, J. Hone, and R. M. Osgood, Phys. Rev. X 3, 021014 (2013).
[CrossRef]

Hsieh, I.-W.

Iorsh, I. V.

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

Jia, X.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Katsnelson, M. I.

M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University, 2012).

Kian Ping, L.

Kivshar, Yu. S.

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

Kockaert, P.

Kong, J.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Kumar, J.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Kumar, N.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Kwong, D. L.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Liu, X.

Lo, G. Q.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Loh, K. P.

Q. Bao and K. P. Loh, ACS Nano 6, 3677 (2012).
[CrossRef]

Marini, A.

H. Dong, C. Conti, A. Marini, and F. Biancalana, J. Phys. B 6, 155401 (2013).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Martin-Moreno, L.

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

Massar, S.

McMillan, J. F.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Michaelis, D.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Mikhailov, S. A.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

S. A. Mikhailov and K. Ziegler, J. Phys. 20, 384204 (2008).
[CrossRef]

S. A. Mikhailov, Europhys. Lett. 79, 27002 (2007).
[CrossRef]

Moger, J.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

Monro, T. M.

Nesterov, M. L.

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

Nezich, D.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Nikitin, A. Yu.

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

Osgood, R. M.

Panoiu, N. C.

Peschel, U.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Petrone, N.

S. Hong, J. I. Dadap, N. Petrone, P. Yeh, J. Hone, and R. M. Osgood, Phys. Rev. X 3, 021014 (2013).
[CrossRef]

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Reina, A.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Robinson, J.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Russell, P. St. J.

Savchenko, A. K.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

Shadrivov, I. V.

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

Skryabin, D. V.

X. Zhao, A. V. Gorbach, and D. V. Skryabin, J. Opt. Soc. Am. B 30, 812 (2013).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Smirl, A. L.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Smirnova, D. A.

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

Son, H.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Sun, Z.

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Tadjerb, M.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

van der Zande, A.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

van Driel, H. V.

J. J. Dean and H. V. van Driel, Phys. Rev. B 82, 125411 (2010).
[CrossRef]

Virally, S.

Vlasov, Y. A.

Wachter, C.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Wadsworth, W. J.

Wang, R.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Wang, S.

S. Chu, S. Wang, and Q. Gong, Chem. Phys. Lett. 523, 104 (2012).
[CrossRef]

Wong, C. W.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Yeh, P.

S. Hong, J. I. Dadap, N. Petrone, P. Yeh, J. Hone, and R. M. Osgood, Phys. Rev. X 3, 021014 (2013).
[CrossRef]

Yu, M.

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

Zhang, H.

Zhao, H.

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Zhao, X.

Ziegler, K.

S. A. Mikhailov and K. Ziegler, J. Phys. 20, 384204 (2008).
[CrossRef]

ACS Nano

Q. Bao and K. P. Loh, ACS Nano 6, 3677 (2012).
[CrossRef]

Adv. Opt. Photon.

Appl. Phys. Lett.

M. Currie, J. D. Caldwell, F. J. Bezares, J. Robinson, T. Anderson, H. Chun, and M. Tadjerb, Appl. Phys. Lett. 99, 211909 (2011).
[CrossRef]

Chem. Phys. Lett.

S. Chu, S. Wang, and Q. Gong, Chem. Phys. Lett. 523, 104 (2012).
[CrossRef]

Europhys. Lett.

S. A. Mikhailov, Europhys. Lett. 79, 27002 (2007).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys.

S. A. Mikhailov and K. Ziegler, J. Phys. 20, 384204 (2008).
[CrossRef]

J. Phys. B

H. Dong, C. Conti, A. Marini, and F. Biancalana, J. Phys. B 6, 155401 (2013).
[CrossRef]

Laser Photon. Rev.

M. L. Nesterov, J. Bravo-Abad, A. Yu. Nikitin, F. J. Garcia-Vidal, and L. Martin-Moreno, Laser Photon. Rev. 7, L7 (2013).

Nano Lett.

A. Reina, X. Jia, J. Z. Ho, D. Nezich, H. Son, V. Bulovic, M. Dresselhaus, and J. Kong, Nano Lett. 9, 30 (2009).
[CrossRef]

Nat. Photonics

T. Gu, N. Petrone, J. F. McMillan, A. van der Zande, M. Yu, G. Q. Lo, D. L. Kwong, J. Hone, and C. W. Wong, Nat. Photonics 6, 554 (2012).
[CrossRef]

F. Bonaccorso, Z. Sun, T. Hasan, and A. C. Ferrari, Nat. Photonics 4, 611 (2010).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

A. V. Gorbach, Phys. Rev. A 87, 013830 (2013).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Phys. Rev. B

D. A. Smirnova, A. V. Gorbach, I. V. Iorsh, I. V. Shadrivov, and Yu. S. Kivshar, Phys. Rev. B 88, 045443 (2013).
[CrossRef]

J. J. Dean and H. V. van Driel, Phys. Rev. B 82, 125411 (2010).
[CrossRef]

N. Kumar, J. Kumar, C. Gerstenkorn, R. Wang, H. Chiu, A. L. Smirl, and H. Zhao, Phys. Rev. B 87, 121406 (2013).
[CrossRef]

Phys. Rev. E

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Phys. Rev. Lett.

E. Hendry, P. J. Hale, J. Moger, A. K. Savchenko, and S. A. Mikhailov, Phys. Rev. Lett. 105, 097401 (2010).
[CrossRef]

Phys. Rev. X

S. Hong, J. I. Dadap, N. Petrone, P. Yeh, J. Hone, and R. M. Osgood, Phys. Rev. X 3, 021014 (2013).
[CrossRef]

Other

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

M. I. Katsnelson, Graphene: Carbon in Two Dimensions (Cambridge University, 2012).

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Figures (4)

Fig. 1.
Fig. 1.

Tapered fiber partially covered with graphene: (a) schematic view, the overlap spans the angle Φ, 0<Φ<2π. (b) Fundamental mode profile of the bare silica glass fiber with core diameter 2R=λ0=1μm.

Fig. 2.
Fig. 2.

Nonlinear coefficient of the fundamental mode of the silica glass fiber with R=0.5μm, χ3=4·1022m2/V2.

Fig. 3.
Fig. 3.

Graphene contribution to the nonlinearity (a) and propagation loss (b). Solid (dashed) curve in (a) corresponds to s˜=1 (s˜=0).

Fig. 4.
Fig. 4.

Power required to match nonlinear and attenuation distances. Solid (dashed) curve corresponds to s˜=1 (s˜=0).

Equations (21)

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E⃗(r⃗,t)=12π+E(r⃗,ω)eiωtdω+c.c.
E(r,ϕ,z,ω)=Aω(z)eω(r,ϕ).
D(ω)=ϵ0ϵE(ω)+ϵ0χ^(3)E(ω1)E*(ω2)E(ω3),
J(ω)=σ^(1)J(ω)+σ^(3)J(ω1)J*(ω2)J(ω3),
izAω=(dω+iαω)Aωω2πΓAω1Aω2*Aω3dω1dω2,
dω=β(ω)+ν(ω).
A(z,t)=12π+Aω(z)ei(ω0+δ)tdδ,
izψ+D^(iτ)ψ+iΛ^(iτ)ψ+γ|ψ|2ψ=0,
D^=k=2Ndkk!(iτ)k,Λ^=k=0Nαkk!(iτ)k,
(ν+iα)=fgπk0Rβ0Piσ1cϵ0Θ[|ez|2+|eϕ|2],
γ=γD+γG,
γD=gπk03β02P2χ3cϵ00Rr[|e⃗|4+12|e⃗2|2]dr,
γG=fgπk03β02P2iσ3(cϵ0)2Rk0Θ[|e⃗1|4|e⃗|2+12e⃗12(e⃗*)2].
P=2π0+r|e⃗|2dr,
g=1/(1+η)2,
η=2πRβ0PΔ[er*ez],
Θ[F(r)]=12limδr0[F(Rδr)+F(R+δr)],
Δ[F(r)]=limδr0[F(Rδr)F(R+δr)].
h=χ3(gr)χ3=σ3cϵ0χ3,
ϵ^g=iσ^(1)cϵ0·cωδ(rRg),
χ^g(3)=iσ^(3)cϵ0·c(ω1ω2ω3)1/3δ(rRg).

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