Abstract

We have investigated the modulation instability (MI) in a nonlinear optical coupler using a generalized model describing the pulse propagation of a waveguiding structure composed of two adjacent waveguides. The model consists of a nonlinear tunnel-coupled structure consisting of right- and left-handed media. The optical coupler considered here includes a local saturable nonlinear refractive index. In particular, we discuss the impact of the saturable nonlinearity for the MI of plane waves and formation of spatial solitons. The results show that MI can exist not only in the normal group velocity dispersion (GVD) regime but also in the normal GVD regime in the nonlinear positive-negative index coupler in the presence of saturable nonlinearity. The saturable nonlinearity can increase/decrease the number of sidebands or shift the existing sidebands. The maximum value of the MI gain, as well as its bandwidth, has been also affected by the saturable nonlinearity. Moreover, the saturable nonlinearity may influence considerably the number of wave trains induced by MI.

© 2012 Optical Society of America

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2012 (3)

R. Schiek, A. S. Solntsev, and D. N. Neshev, “Temporal dynamics of all-optical switching in quadratic nonlinear directional couplers,” Appl. Phys. Lett. 100, 111117 (2012).
[CrossRef]

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

Z. Kudyshev, G. Venugopal, and N. M. Litchinitser, “Generalized analytical solutions for nonlinear positive-negative index couplers,” Phys. Res. Int. 2012, 945807 (2012).
[CrossRef]

2011 (2)

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

X. Zhong, T. Tang, A. Xiang, and K. Cheng, “Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects,” Opt. Commun. 284, 4727–4731 (2011).
[CrossRef]

2010 (4)

R. V. J. Raja, K. Porsezian, and K. Nithyanandan, “Modulational-instability-induced supercontinuum generation with saturable nonlinear response,” Phys. Rev. A 82, 013825 (2010).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulation instability in nonlinear oppositely directed coupler with a negative-index metamaterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

P. T. Dinda and K. Porsezian, “Impact of fourth-order dispersion in the modulational instability spectra of wave propagation in glass fibers with saturable nonlinearity,” J. Opt. Soc. Am. B 27, 1143–1152 (2010).
[CrossRef]

A. Mohamadou, C. G. LatchioTiofack, and T. C. Kofane, “Wave train generation of solitons in systems with higher-order nonlinearities,” Phys. Rev. E 82, 016601 (2010).
[CrossRef]

2008 (2)

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, “Solitary waves in a nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

A. Maluckov, L. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

2007 (1)

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

2006 (4)

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

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, and C. M. Soukoulis, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

V. M. Agranovich, and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Phys. Uspekhi 49, 1029–1044 (2006).
[CrossRef]

2005 (2)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[CrossRef]

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

2004 (4)

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

2003 (1)

2001 (2)

R. A. Shelby, D. R. Smith, S. N. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

2000 (1)

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Young, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

1994 (2)

1991 (2)

1988 (2)

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

V. E. Wood, E. D. Evan, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 69, 156–160(1988).
[CrossRef]

1987 (2)

G. I. Stegeman, C. T. Seaton, C. N. Ironside, and T. Cullen, “Effects of saturation and loss on nonlinear directional couplers,” Appl. Phys. Lett. 50, 1035–1037 (1987).
[CrossRef]

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

1986 (1)

R. Hoffe and J. Chrostowki, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

1985 (3)

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

U. Langbein, F. Lederer, T. Peschel, and H. -E. Ponath, “Nonlinear guided waves in saturable nonlinear media,” Opt. Lett. 10, 571–573 (1985).
[CrossRef]

1982 (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE. J. Quantum Electron. 18, 1580–1583 (1982).
[CrossRef]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich, and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Phys. Uspekhi 49, 1029–1044 (2006).
[CrossRef]

Agrawal, G. P.

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

Alim,

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

Alu, A.

A. Alu and N. Engheta, “An overview of salient properties of planar guided-wave structures with double-negative (DNG) and single-negative (SNG) layers,” in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G. V. Eleftheriades and K. G. Balmain, eds. (John Wiley and Sons Inc., 2005), pp. 339–380.

Anand, S.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Andrejco, M. J.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

Au Kong, J.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Berrier, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Brueck, S. R.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Chen, H.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Cheng, K.

X. Zhong, T. Tang, A. Xiang, and K. Cheng, “Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects,” Opt. Commun. 284, 4727–4731 (2011).
[CrossRef]

Chrostowki, J.

R. Hoffe and J. Chrostowki, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

Cullen, T.

G. I. Stegeman, C. T. Seaton, C. N. Ironside, and T. Cullen, “Effects of saturation and loss on nonlinear directional couplers,” Appl. Phys. Lett. 50, 1035–1037 (1987).
[CrossRef]

Dai, X.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulation instability in nonlinear oppositely directed coupler with a negative-index metamaterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

Dianov, E. M.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Dinda, P. T.

Dolling, G.

Elson, J. M.

Engheta, N.

A. Alu and N. Engheta, “An overview of salient properties of planar guided-wave structures with double-negative (DNG) and single-negative (SNG) layers,” in Negative-Refraction Metamaterials: Fundamental Principles and Applications, G. V. Eleftheriades and K. G. Balmain, eds. (John Wiley and Sons Inc., 2005), pp. 339–380.

Enkrich, C.

G. Dolling, C. Enkrich, M. Wegener, and C. M. Soukoulis, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Evan, E. D.

V. E. Wood, E. D. Evan, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 69, 156–160(1988).
[CrossRef]

Fan, D.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulation instability in nonlinear oppositely directed coupler with a negative-index metamaterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

Fan, W.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Friberg, S. R.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

Fu, X.

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

Gabitov, I. R.

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, “Solitary waves in a nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Gartstein, Y. N.

V. M. Agranovich, and Y. N. Gartstein, “Spatial dispersion and negative refraction of light,” Phys. Uspekhi 49, 1029–1044 (2006).
[CrossRef]

Gatz, S.

Grzegorczyk, T. M.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Gusovskii, D. D.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Hadzievski, L.

A. Maluckov, L. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

Halterman, K.

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, 1990).

Heatley, D. R.

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

Herrmann, J.

Hoffe, R.

R. Hoffe and J. Chrostowki, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Young, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Huangfu, J.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Ironside, C. N.

G. I. Stegeman, C. T. Seaton, C. N. Ironside, and T. Cullen, “Effects of saturation and loss on nonlinear directional couplers,” Appl. Phys. Lett. 50, 1035–1037 (1987).
[CrossRef]

Jensen, S. M.

S. M. Jensen, “The nonlinear coherent coupler,” IEEE. J. Quantum Electron. 18, 1580–1583 (1982).
[CrossRef]

Kenan, R. P.

V. E. Wood, E. D. Evan, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 69, 156–160(1988).
[CrossRef]

Kofane, T. C.

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

A. Mohamadou, C. G. LatchioTiofack, and T. C. Kofane, “Wave train generation of solitons in systems with higher-order nonlinearities,” Phys. Rev. E 82, 016601 (2010).
[CrossRef]

Konar, S.

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Kroll, N.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

Kudyshev, Z.

Z. Kudyshev, G. Venugopal, and N. M. Litchinitser, “Generalized analytical solutions for nonlinear positive-negative index couplers,” Phys. Res. Int. 2012, 945807 (2012).
[CrossRef]

Langbein, U.

LatchioTiofack, C. G.

A. Mohamadou, C. G. LatchioTiofack, and T. C. Kofane, “Wave train generation of solitons in systems with higher-order nonlinearities,” Phys. Rev. E 82, 016601 (2010).
[CrossRef]

Lazarides, N.

A. Maluckov, L. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

Lederer, F.

Li Kam Wa, P.

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

Linden, S.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Litchinitser, N. M.

Z. Kudyshev, G. Venugopal, and N. M. Litchinitser, “Generalized analytical solutions for nonlinear positive-negative index couplers,” Phys. Res. Int. 2012, 945807 (2012).
[CrossRef]

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, “Solitary waves in a nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

Maier, A. A.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Maimistov, A. I.

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, “Solitary waves in a nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

N. M. Litchinitser, I. R. Gabitov, and A. I. Maimistov, “Optical bistability in a nonlinear optical coupler with a negative index channel,” Phys. Rev. Lett. 99, 113902 (2007).
[CrossRef]

Malloy, K. J.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Maluckov, A.

A. Maluckov, L. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

Mason, N. J.

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

Medhekar, S.

Mohamadou, A.

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

A. Mohamadou, C. G. LatchioTiofack, and T. C. Kofane, “Wave train generation of solitons in systems with higher-order nonlinearities,” Phys. Rev. E 82, 016601 (2010).
[CrossRef]

Mulot, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Nemat-Nasser, S. N.

R. A. Shelby, D. R. Smith, S. N. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

Neshev, D. N.

R. Schiek, A. S. Solntsev, and D. N. Neshev, “Temporal dynamics of all-optical switching in quadratic nonlinear directional couplers,” Appl. Phys. Lett. 100, 111117 (2012).
[CrossRef]

Neustruev, V. B.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Nithyanandan, K.

R. V. J. Raja, K. Porsezian, and K. Nithyanandan, “Modulational-instability-induced supercontinuum generation with saturable nonlinear response,” Phys. Rev. A 82, 013825 (2010).
[CrossRef]

Oliver, M. K.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

Osgood, R. M.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Overfelt, P. L.

Panoiu, N. C.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

Pendry, J. B.

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Young, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Peschel, T.

Ponath, H. -E.

Porsezian, K.

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

R. V. J. Raja, K. Porsezian, and K. Nithyanandan, “Modulational-instability-induced supercontinuum generation with saturable nonlinear response,” Phys. Rev. A 82, 013825 (2010).
[CrossRef]

P. T. Dinda and K. Porsezian, “Impact of fourth-order dispersion in the modulational instability spectra of wave propagation in glass fibers with saturable nonlinearity,” J. Opt. Soc. Am. B 27, 1143–1152 (2010).
[CrossRef]

Qiu, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Raja, R. V. J.

R. V. J. Raja, K. Porsezian, and K. Nithyanandan, “Modulational-instability-induced supercontinuum generation with saturable nonlinear response,” Phys. Rev. A 82, 013825 (2010).
[CrossRef]

Rajkamal,

Ramakrishna, S. A.

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68, 449–521 (2005).
[CrossRef]

Ran, L.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Roberts, J. S.

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

Robson, P. N.

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

Saifi, M. A.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

Saxena, A.

Schiek, R.

R. Schiek, A. S. Solntsev, and D. N. Neshev, “Temporal dynamics of all-optical switching in quadratic nonlinear directional couplers,” Appl. Phys. Lett. 100, 111117 (2012).
[CrossRef]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, S. N. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

Seaton, C. T.

G. I. Stegeman, C. T. Seaton, C. N. Ironside, and T. Cullen, “Effects of saturation and loss on nonlinear directional couplers,” Appl. Phys. Lett. 50, 1035–1037 (1987).
[CrossRef]

Sengupta, A.

Shcherbakov, I. A.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, S. N. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

Shklovskii, E. I.

D. D. Gusovskii, E. M. Dianov, A. A. Maier, V. B. Neustruev, E. I. Shklovskii, and I. A. Shcherbakov, “Nonlinear light transfer in tunnel-coupled optical waveguides,” Sov. J. Quantum Electron. 15, 1523–1526 (1985).
[CrossRef]

Silberberg, Y.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

Sitch, J. E.

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

Smith, D. R.

R. A. Shelby, D. R. Smith, S. N. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000).
[CrossRef]

Smith, P. W.

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

Sodha, M. S.

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

Solntsev, A. S.

R. Schiek, A. S. Solntsev, and D. N. Neshev, “Temporal dynamics of all-optical switching in quadratic nonlinear directional couplers,” Appl. Phys. Lett. 100, 111117 (2012).
[CrossRef]

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, and C. M. Soukoulis, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

Stegeman, G. I.

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

G. I. Stegeman, C. T. Seaton, C. N. Ironside, and T. Cullen, “Effects of saturation and loss on nonlinear directional couplers,” Appl. Phys. Lett. 50, 1035–1037 (1987).
[CrossRef]

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Young, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Su,

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

Swillo, M.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Talneau, A.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Tang, T.

X. Zhong, T. Tang, A. Xiang, and K. Cheng, “Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects,” Opt. Commun. 284, 4727–4731 (2011).
[CrossRef]

Thylen, L.

A. Berrier, M. Mulot, M. Swillo, M. Qiu, L. Thylen, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902 (2004).
[CrossRef]

Tiofack, C. G. L.

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

Tsironis, G. P.

A. Maluckov, L. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

Venugopal, G.

Z. Kudyshev, G. Venugopal, and N. M. Litchinitser, “Generalized analytical solutions for nonlinear positive-negative index couplers,” Phys. Res. Int. 2012, 945807 (2012).
[CrossRef]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401 (2004).
[CrossRef]

Wang, Y.

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

Wegener, M.

G. Dolling, C. Enkrich, M. Wegener, and C. M. Soukoulis, “Low-loss negative-index metamaterial at telecommunication wavelengths,” Opt. Lett. 31, 1800–1802 (2006).
[CrossRef]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Wen, S.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulation instability in nonlinear oppositely directed coupler with a negative-index metamaterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

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

Wood, V. E.

V. E. Wood, E. D. Evan, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 69, 156–160(1988).
[CrossRef]

Wright, E. M.

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

Xiang, A.

X. Zhong, T. Tang, A. Xiang, and K. Cheng, “Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects,” Opt. Commun. 284, 4727–4731 (2011).
[CrossRef]

Xiang, Y.

Y. Xiang, X. Dai, S. Wen, and D. Fan, “Modulation instability in metamaterials with saturable nonlinearity,” J. Opt. Soc. Am. B 28, 908–916 (2011).
[CrossRef]

Y. Xiang, S. Wen, X. Dai, and D. Fan, “Modulation instability in nonlinear oppositely directed coupler with a negative-index metamaterial channel,” Phys. Rev. E 82, 056605 (2010).
[CrossRef]

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

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984).

Young, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Young, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996).
[CrossRef]

Yuan, Y.

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

Zhang, S.

S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).
[CrossRef]

Zhong, X.

X. Zhong, T. Tang, A. Xiang, and K. Cheng, “Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects,” Opt. Commun. 284, 4727–4731 (2011).
[CrossRef]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004).
[CrossRef]

Appl. Phys. Lett. (7)

R. A. Shelby, D. R. Smith, S. N. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489–491 (2001).
[CrossRef]

S. R. Friberg, Y. Silberberg, M. K. Oliver, M. J. Andrejco, M. A. Saifi, and P. W. Smith, “Ultrafast all-optical switching in a dual-core fiber nonlinear coupler,” Appl. Phys. Lett. 51, 1135–1137 (1987).
[CrossRef]

D. R. Heatley, E. M. Wright, and G. I. Stegeman, “Soliton coupler,” Appl. Phys. Lett. 53, 172–174 (1988).
[CrossRef]

P. Li Kam Wa, J. E. Sitch, N. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple-quantum-well waveguide switch,” Appl. Phys. Lett. 21, 26–28 (1985).

Y. Yuan, L. Ran, H. Chen, J. Huangfu, T. M. Grzegorczyk, and J. Au Kong, “Backward coupling waveguide coupler using left-handed material,” Appl. Phys. Lett. 88, 211903 (2006).
[CrossRef]

R. Schiek, A. S. Solntsev, and D. N. Neshev, “Temporal dynamics of all-optical switching in quadratic nonlinear directional couplers,” Appl. Phys. Lett. 100, 111117 (2012).
[CrossRef]

G. I. Stegeman, C. T. Seaton, C. N. Ironside, and T. Cullen, “Effects of saturation and loss on nonlinear directional couplers,” Appl. Phys. Lett. 50, 1035–1037 (1987).
[CrossRef]

Contemp. Phys. (1)

J. B. Pendry, “Negative refraction,” Contemp. Phys. 45, 191–202 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9, 919–933 (1973).
[CrossRef]

IEEE. J. Quantum Electron. (1)

S. M. Jensen, “The nonlinear coherent coupler,” IEEE. J. Quantum Electron. 18, 1580–1583 (1982).
[CrossRef]

J. Mod. Opt. (1)

C. G. L. Tiofack, A. Mohamadou, Alim, K. Porsezian, and T. C. Kofane, “Modulational instability in metamaterials with saturable nonlinearity and higher-order dispersion,” J. Mod. Opt. 59, 972–979 (2012).
[CrossRef]

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

Opt. Commun. (3)

X. Zhong, T. Tang, A. Xiang, and K. Cheng, “Modulation instability in negative refractive metamaterials with exponential saturable nonlinearity and self-steepening effects,” Opt. Commun. 284, 4727–4731 (2011).
[CrossRef]

V. E. Wood, E. D. Evan, and R. P. Kenan, “Soluble saturable refractive-index model,” Opt. Commun. 69, 156–160(1988).
[CrossRef]

R. Hoffe and J. Chrostowki, “Optical pulse compression and breaking in nonlinear fibre couplers,” Opt. Commun. 57, 34–38 (1986).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Opt. Spectrosc. (1)

A. I. Maimistov, I. R. Gabitov, and N. M. Litchinitser, “Solitary waves in a nonlinear oppositely directed coupler,” Opt. Spectrosc. 104, 253–257 (2008).
[CrossRef]

Phys. Res. Int. (1)

Z. Kudyshev, G. Venugopal, and N. M. Litchinitser, “Generalized analytical solutions for nonlinear positive-negative index couplers,” Phys. Res. Int. 2012, 945807 (2012).
[CrossRef]

Phys. Rev. A (1)

R. V. J. Raja, K. Porsezian, and K. Nithyanandan, “Modulational-instability-induced supercontinuum generation with saturable nonlinear response,” Phys. Rev. A 82, 013825 (2010).
[CrossRef]

Phys. Rev. E (4)

A. Maluckov, L. Hadzievski, N. Lazarides, and G. P. Tsironis, “Left-handed metamaterials with saturable nonlinearity,” Phys. Rev. E 77, 046607 (2008).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Gain spectrum versus f (f>0) and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) ϒ1=1, ϒ2=0, λ=0.9 and (b) ϒ1=0, ϒ2=1, λ=0.015.

Fig. 2.
Fig. 2.

Gain spectrum versus f (f>0) and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) ϒ1=1, ϒ2=1, λ=0.015 and (b) ϒ1=1, ϒ2=1, λ=0.015.

Fig. 3.
Fig. 3.

Gain spectrum versus f (f<0) and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) ϒ1=1, ϒ2=0, λ=0.04 and (b) ϒ1=0, ϒ2=1, λ=0.01.

Fig. 4.
Fig. 4.

Gain spectrum versus f (f<0) and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) ϒ1=1, ϒ2=1, λ=0.025 and (b) ϒ1=1, ϒ2=1, λ=0.010.

Fig. 5.
Fig. 5.

Gain spectrum versus ϒ2 and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) Normal dispersion regime f=1, ϒ1=1, λ=0.004 and (b) f=1, ϒ1=1, λ=0.015.

Fig. 6.
Fig. 6.

Gain spectrum versus ϒ2 and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) Anomalous dispersion regime f=1, ϒ1=1, λ=0.15 and (b) f=1, ϒ1=1, λ=0.04, with negative ϒ2, λ=0.02.

Fig. 7.
Fig. 7.

Gain spectral versus power P and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) ϒ1=1, ϒ2=1, λ=0.02, f=0.7 and (b) ϒ1=1, ϒ2=1, λ=0.02, f=0.7.

Fig. 8.
Fig. 8.

Gain spectral versus power P and wave vector K under different nonlinear conditions for k=10cm1 and P=10cm1. (a) ϒ1=1, ϒ2=1, λ=0.02, f=0.9 and (b) ϒ1=1, ϒ2=1, λ=0.0015, f=0.9.

Equations (22)

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iσ1a1z+i1v1ga1t+k12a2exp(iδz)+γ1|a1|2a1=0,
iσ2a2z+i1v2ga2t+k21a1exp(iδz)+γ2|a2|2a2=0,
f(λ|ai|2)=λ|ai|21+λ|ai|2,
iσ1a1z+i1v1ga1t+k12a2exp(iδz)+γ1|a1|2a11+λ|a1|2=0,
iσ2a2z+i1v2ga2t+k21a1exp(iδz)+γ2|a2|2a21+λ|a2|2=0.
δ=k12f2+k21fP1+f2(γ1+γ2f2)P2(1+f2)2λ(γ1+γ2f4),
q=k21k12f22fP2(1+f2)(γ2f2γ1)P22(1+f2)2λ(γ2f4γ1),
ib1z+i1v1gb1t+k12b2k12fb1+P1+f2[γ1(b1+b1*)]2P2(1+f2)2λ[γ1(b1+b1*)]γ1λ(P1+f2)3/2b1=0,
ib2z+i1v2gb2t+k21b1k21f1b2+P1+f2[γ2f2(b2+b2*)]2P2(1+f2)2λ[γ2f4(b2+b2*)]γ2λ(P1+f2)3/2b2=0.
bj=cjexp[i(KzΩt)]+djexp[i(KzΩt)],
(L11L12L13L14L21L22L23L24L31L32L33L34L41L42L43L44)(c1c2d1d2)=0,
L11=K+S1k12f+G1λG3λG5,L12=k12,L13=G1λG3,L14=0,L21=k21,L22=K+S2+G2f2k21f1λG4f4λG6,L23=0,L24=G2f2λG4f4,L31=G1λG3,L32=0;L33=KS1k12f+G1λG3λG5,L34=k12,L41=0,L42=G2f2λG4f4,L43=k21,L44=KS2k21f1+G2f2λG4f4λG6,
G1=P1+f2γ1,G2=P1+f2γ2,G3=2P2(1+f2)2γ1,G4=2P2(1+f2)2γ2,G5=(P1+f2)3/2γ1,G6=(P1+f2)3/2γ2.
S4AS2+DS+C=0,
S=±12ϒ3±1263Aϒ32D/ϒ3,
ϒ1=2A3+27D2+72AC,
ϒ2=[ϒ1+4(A2+12C)3+ϒ12]1/3/21/3,
ϒ3=23A+(A2+12C)/(3ϒ2)+13ϒ2.
G=|Im(SMax)|,
A=k2f2+2f4G4G6λ2+2fkλG32kfG1+2G3G5λ2+2k2+k2f2+G52λ+2K2+2kfG5λ2G1G5λ+G62λ2f2G2G6λ+2f·k3G4λ+2kG6λf2fkG2,
D=4kfKG5λ4kfKG3λ+4kfKG12Kk2f24G3KG5λ2+4f3KkG4λ+4f4KG4G6λ24f2KG2G6λ+2KG62λ+2Kk2f24fKkG22KG52λ+4G1KG5λ+4KkG6λf,
C=K2k2f22G5k3λf+G62G52λ2K2G52λ+G52k2λf22G3k3λf+G62kf2λK2k2f2+2K2k2+2k3G3λf+2k3G5λf2G1G5k2λf2+2G3G5k2λ2f2+2fkK2G22f3kK2G4λ2kK2G6λf+2f2G2K2G6λ2f4G4K2G6λ2+2k2fG6λ+2f5k2G4λ2f4G2kG6λ+2f6G4kG6λ22kfK2G3λ+2kfK2G12kfK2G5λ2f5k3G4λ+2kfG62G5λ22kfG62G1λ+2kfG62G3λ22k3fG6λ+2G1K2G5λ+4f4G1G4k2λ2G1G62G5λ2+2G1G6k2λ2G3K2G5λ2+4f2G3G2k2λ4f4G3G4k2λ2+2G3G62G5λ32G3G6k2λ22fkG52G2λ+2kG52G6λ2f+2f3kG52G4λ22f2G2G52G6λ2+2f4G4G52G6λ3+2f2G5G2k2λ2f4G5G4k2λ22G5G6k2λ2+K4G62K2λ4f4k2G1G4λ4f2k2G5G2λ+4k2G5G6λ24k2G1G6λ+4f4k2G3G4λ2+4f4k2G5G4λ2+4k2G3G6λ24f2k2G3G2λ4f3kG2λ2G5G6+4f3kG2G1G6λ4f3kG2G3G6λ2+4f5kG4G5G6λ34f5kG4G1G6λ2+4f5kG4G3G6λ3+4fG1kG5G2λ4G1kG5G6λ2f4f3G1kG5G4λ2+4f2G1G2G5G6λ24f4G1G4G5G6λ34fG3kG5G2λ2+4G3kG5G6λ3f+4f3G3kG5G4λ34f2G3G2G5G6λ3+4f4G3G4G5G6λ4.

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