Abstract

Linear stability analysis predicts that transverse modulation instability (MI) can occur for two optical beams copropagating in a Kerr nonlinear metamaterial (MM) interacting with each other through cross-phase modulation. A detailed discussion on the role of cross-phase modulation-induced coupling between beams on MI is presented, and it is found that MMs will enrich the propagating characteristics of beams. MI that occurs when beams are in different refractive-index regions is particularly interesting since it is a precursor to the formation of bright and dark spatial solitons. Furthermore, transverse MI can develop for different combinations of the signs of refractive index decided by the two beams, respectively, which makes the temporal MI in optical fibers find its transverse MI counterpart in MMs. Based on the Drude model, numerical simulations are performed to confirm theoretical predictions.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
  27. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
    [CrossRef]

2007 (4)

G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53-55 (2007).
[CrossRef]

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058-3063 (2007).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
[CrossRef]

2006 (4)

2005 (4)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

V. M. Shalaev, W. Cai, U. K. Chettiar, H.-K. Yuan, A. K. Sarychev, V. P. Drachev, and A. V. Kildishev, “Negative index of refraction in optical metamaterials,” Opt. Lett. 30, 3356-3358 (2005).
[CrossRef]

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

2004 (2)

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

2003 (2)

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

2002 (1)

2001 (1)

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

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

1997 (1)

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]

1992 (1)

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202-4208 (1992).
[CrossRef] [PubMed]

1990 (2)

G. P. Agrawal, “Transverse modulation instability of copropagating optical beams in nonlinear Kerr media,” J. Opt. Soc. Am. B 7, 1072-1078 (1990).
[CrossRef]

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414-442 (1990).
[CrossRef]

1989 (1)

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406-3413 (1989).
[CrossRef] [PubMed]

1988 (1)

1987 (1)

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Abbi, S. C.

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414-442 (1990).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Agrawal, G. P.

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

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202-4208 (1992).
[CrossRef] [PubMed]

G. P. Agrawal, “Transverse modulation instability of copropagating optical beams in nonlinear Kerr media,” J. Opt. Soc. Am. B 7, 1072-1078 (1990).
[CrossRef]

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406-3413 (1989).
[CrossRef] [PubMed]

Akozbek, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Alfano, R. R.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406-3413 (1989).
[CrossRef] [PubMed]

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]

Baldeck, P. L.

G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, “Modulation instability induced by cross-phase modulation in optical fibers,” Phys. Rev. A 39, 3406-3413 (1989).
[CrossRef] [PubMed]

Baughman, R. H.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Cai, W.

Cao, X. D.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202-4208 (1992).
[CrossRef] [PubMed]

Chettiar, U. K.

D'Aguanno, G.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Dai, X.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058-3063 (2007).
[CrossRef]

Dolling, G.

Drachev, V. P.

Enkrich, C.

Fan, D.

Firth, W. J.

Fu, X.

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

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]

Gorkunov, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Hu, Y.

Kildishev, A. V.

Kivshar, Y. S.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Kothari, N. C.

N. C. Kothari and S. C. Abbi, “Instability growth and filamentation of very intense laser beams in self-focusing media,” Prog. Theor. Phys. 83, 414-442 (1990).
[CrossRef]

Kourakis, I.

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Lapine, M.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[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]

Lazarides, N.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Linden, S.

Liou, L. W.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202-4208 (1992).
[CrossRef] [PubMed]

Mattiucci, N.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

McKinstrie, C. J.

L. W. Liou, X. D. Cao, C. J. McKinstrie, and G. P. Agrawal, “Spatiotemporal instabilities in dispersive nonlinear media,” Phys. Rev. A 46, 4202-4208 (1992).
[CrossRef] [PubMed]

Paré, C.

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Poliakov, E. Y.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Popov, A. K.

Potosek, M. J.

Ringhofer, K. H.

M. Lapine, M. Gorkunov, and K. H. Ringhofer, “Nonlinearity of a metamaterial arising from diode insertions into resonant conductive elements,” Phys. Rev. E 67, 065601 (2003).
[CrossRef]

Sarychev, A. K.

Scalora, M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Shadrivov, I. V.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Shalaev, V. M.

Shen, Y. R.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Shukla, P. K.

I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E 72, 016626 (2005).
[CrossRef]

Soukoulis, C. M.

Stegeman, G. I.

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]

Su, W.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058-3063 (2007).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Syrchin, M. S.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Tang, Z.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058-3063 (2007).
[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]

Trillo, 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]

Tsironis, G. P.

N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E 71, 036614 (2005).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

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, Y.

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

Wegener, M.

Wen, S.

Xiang, Y.

S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A 75, 033815 (2007).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, Z. Tang, W. Su, and D. Fan, “Modulation instability induced by nonlinear dispersion in nonlinear metamaterials,” J. Opt. Soc. Am. B 24, 3058-3063 (2007).
[CrossRef]

S. Wen, Y. Wang, W. Su, Y. Xiang, X. Fu, and D. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E 73, 036617 (2006).
[CrossRef]

S. Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express 14, 1568-1575 (2006).
[CrossRef] [PubMed]

Yuan, H.-K.

Zakhidov, A. A.

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, Phys. Rev. B 69, 165112 (2004).
[CrossRef]

V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B 69, 165112 (2004).
[CrossRef]

Zharov, A. A.

A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003).
[CrossRef] [PubMed]

Zheltikov, A. M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials,” Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

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

Nat. Photonics (1)

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41-48 (2007).
[CrossRef]

Opt. Express (1)

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Figures (6)

Fig. 1
Fig. 1

Refractive index n, wave number k j , and nonlinear coefficient γ j versus ω ω p e for (a) ϖ p = 0.8 , (b) ϖ p = 1 , and (c) ϖ p = 1.2 . k j and γ j are calculated in units of ω pe c and c μ 0 χ ω p e 2 , respectively.

Fig. 2
Fig. 2

MI gain spectrum in the negative-index region of MM with self-focusing nonlinearity for (a) ω pe = 1.36 × 10 16 Hz and (b) for different ω pe .

Fig. 3
Fig. 3

MI gain spectra in the negative-index region of self-defocusing MM for (a) ω pe = 1.36 × 10 16 Hz , where line 1 is for the cases of both beams propagating in the MM, and lines 2 and 3 are for the cases of beam 1 and beam 2 propagating in the MM alone, respectively; (b) for different ω pe .

Fig. 4
Fig. 4

MI gain spectra in self-focusing MM for (a) ω pe = 0.4 × 10 16 Hz , where line 1 is for the case of both beams propagation in the MM and line 2 is for the case of beam 2 propagation in MM alone; (b) for different ω pe .

Fig. 5
Fig. 5

MI gain spectra in self-defocusing MM for (a) ω pe = 0.4 × 10 16 Hz , where line 1 is the case of both beams propagation in MM and line 2 is the case of beam 1 propagation in the MM alone; (b) for different ω pe .

Fig. 6
Fig. 6

Beam profiles at different propagation distances in self-focusing MMs for (a),(b) beam 1 and beam 2 propagation without the XPM, respectively; (c),(d) beam 1 and beam 2 propagation under XPM, respectively.

Equations (19)

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E j = 1 2 A j ( r ) exp [ i ( k j z ω j t ) ] + c.c ,
P NL j = χ ( E j 2 + 2 E 3 j 2 ) E j ,
( 2 z 2 + 2 ) E ( E ) μ ɛ 2 E t 2 = μ 2 P NL t 2 ,
A j z = i 2 k j 2 A j + i γ j ( A j 2 + 2 A 3 j 2 ) A j ,
γ j = χ ω j 2 μ ( ω j ) 2 k j .
U j Z = i 2 δ j 2 U j + i sgn ( χ j ) σ j ( U j 2 + 2 U 3 j 2 ) U j , j = 1 , 2 ,
ɛ ( ω ) = ɛ 0 ( 1 ω p e 2 ω ( ω + i γ e ) ) , μ ( ω ) = μ 0 ( 1 ω p m 2 ω ( ω + i γ m ) ) ,
k j = ω p e ϖ j c ( 1 1 ϖ j 2 ) ( 1 ϖ p 2 ϖ j 2 ) ,
γ j = c μ 0 ω p e χ ϖ j 2 ϖ j 2 ϖ p 2 ϖ j 2 1 ,
U ¯ j ( Z ) = I j exp [ i sgn ( χ j ) σ j ( I j + 2 I 3 j ) Z ] ,
U j = U ¯ j ( z ) { 1 + u j exp [ i ( q X X + q Y Y + K Z ) ] + i v j exp [ i ( q X X + q Y Y + K Z ) ] } ,
( K 2 f 1 ) ( K 2 f 2 ) = C XPM ,
f j = δ j 2 q 2 ( δ j 2 q 2 2 sgn ( χ j ) σ j I j ) ,
C XPM = 4 δ 1 δ 2 sgn ( χ 1 ) σ 1 sgn ( χ 2 ) σ 2 I 1 I 2 q 4 .
K = ± { f 1 + f 2 2 ± [ ( f 1 f 2 ) 2 4 + C XPM ] 1 2 } 1 2 .
( δ 1 4 q 2 2 sgn ( χ 1 ) σ 1 I 1 ) ( δ 2 4 q 2 2 sgn ( χ 2 ) σ 2 I 2 ) < 4 sgn ( χ 1 ) σ 1 sgn ( χ 2 ) σ 2 I 1 I 2 ,
[ δ 1 2 ( δ 1 2 q 2 2 sgn ( χ 1 ) σ 1 I 1 ) δ 2 2 ( δ 2 2 q 2 2 sgn ( χ 2 ) σ 2 I 2 ) ] 2 < 16 δ 1 δ 2 sgn ( χ 1 ) σ 1 sgn ( χ 2 ) σ 2 I 1 I 2 .
g = 2 Im ( K ) = 2 { f 1 + f 2 2 + [ ( f 1 f 2 ) 2 4 + C XPM ] 1 2 } 1 2 ,
g = 2 Im ( K ) = 2 [ ( f 1 f 2 C XPM ) 1 2 ( f 1 + f 2 2 ) ] 1 2 .

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