Abstract

The combination of right-handed and left-handed materials offers the possibility to design devices in which the mean diffraction is zero. Such systems are encountered, for example, in nonlinear optical cavities, where a true zero-diffraction regime could lead to the formation of patterns with arbitrarily small sizes. In practice, the minimal size is limited by nonlocal terms in the equation of propagation. We study the nonlocal properties of light propagation in a nonlinear optical cavity containing a right-handed and a left-handed material. We obtain a model for the propagation, including two sources of nonlocality: the spatial dispersion of the materials in the cavity, and the higher-order terms of the mean field approximation. We apply these results to a particular case and derive an expression for the parameter fixing the minimal size of the patterns.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (6)

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
[CrossRef]

A. D. Boardman, P. Egan, R. C. Mitchell-Thomas, Y. G. Rapoport, and N. J. King, “Weakly and strongly nonlinear waves in negative phase metamaterials,” Proc. SPIE 7029, 70291F (2008).
[CrossRef]

K. Cho, “A single susceptibility scheme of macroscopic Maxwell equations: beyond the 'E, D, B, H' approach,” J. Phys. Condens. Matter 20, 175202 (2008).
[CrossRef]

D. Sjöberg, “A modified Drude-Born-Fedorov model for isotropic chiral media, obtained by finite scale homogenization,” J. Phys. D 41, 155412 (2008).
[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]

A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, “Single-negative, double-negative and low index metamaterials and their electromagnetic applications,” IEEE Trans. Antennas Propag. 49, 23-36 (2007).

P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
[CrossRef] [PubMed]

L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
[CrossRef]

2006 (6)

P. Kockaert, P. Tassin, G. V. der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529-534 (2006).
[CrossRef]

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777-1780 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

K. Staliunas, C. Serrat, R. Herrero, C. Cojocaru, and J. Trull, “Subdiffractive light pulses in photonic crystals,” Phys. Rev. E 74, 016605 (2006).
[CrossRef]

T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
[CrossRef]

2005 (4)

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (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]

D. Sjöberg, “Homogenization of dispersive material parameters for Maxwell's equations using a singular value decomposition,” Multiscale Model. Simul. 4, 760-789 (2005).
[CrossRef]

2004 (4)

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

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]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 6, 788-792 (2004).
[CrossRef]

K. Aydin, M. Guven, L. Kafesaki, C. Zhang, M. Soukoulis, and M. Ozbay, “Experimental observation of true left-handed transmission peaks in metamaterials,” Opt. Lett. 29, 2623-2625 (2004).
[CrossRef] [PubMed]

2003 (3)

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

K. Staliunas, “Midband dissipative spatial solitons,” Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

2002 (1)

N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
[CrossRef]

2001 (2)

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

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

2000 (3)

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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

A. D. Boardman, K. Marinov, D. I. Pushkarov, and A. Shivarova, “Influence of nonlinearly induced diffraction on spatial solitary waves,” Opt. Quantum Electron. 32, 49-62 (2000).
[CrossRef]

1999 (1)

A. P. Vinogradov and A. V. Aivazyan, “Scaling theory for homogenization of the Maxwell equations,” Phys. Rev. E 60, 987-993 (1999).
[CrossRef]

1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ϵ and μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[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 and V. L. Ginzburg, “Crystal optics with spatial dispersion” in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. IX, Chap. 6, pp. 235-280.
[CrossRef]

Aivazyan, A. V.

A. P. Vinogradov and A. V. Aivazyan, “Scaling theory for homogenization of the Maxwell equations,” Phys. Rev. E 60, 987-993 (1999).
[CrossRef]

Alu, A.

A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, “Single-negative, double-negative and low index metamaterials and their electromagnetic applications,” IEEE Trans. Antennas Propag. 49, 23-36 (2007).

Aydin, K.

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]

Bloemer, M. J.

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Boardman, A. D.

A. D. Boardman, P. Egan, R. C. Mitchell-Thomas, Y. G. Rapoport, and N. J. King, “Weakly and strongly nonlinear waves in negative phase metamaterials,” Proc. SPIE 7029, 70291F (2008).
[CrossRef]

A. D. Boardman, K. Marinov, D. I. Pushkarov, and A. Shivarova, “Influence of nonlinearly induced diffraction on spatial solitary waves,” Opt. Quantum Electron. 32, 49-62 (2000).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, 1992).

Brock, J. B.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Cho, K.

K. Cho, “A single susceptibility scheme of macroscopic Maxwell equations: beyond the 'E, D, B, H' approach,” J. Phys. Condens. Matter 20, 175202 (2008).
[CrossRef]

Chuang, I. L.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Cojocaru, C.

K. Staliunas, C. Serrat, R. Herrero, C. Cojocaru, and J. Trull, “Subdiffractive light pulses in photonic crystals,” Phys. Rev. E 74, 016605 (2006).
[CrossRef]

Colet, P.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
[CrossRef]

Cummer, S. A.

M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

D'Aguanno, G.

G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Danckaert, J.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
[CrossRef]

L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
[CrossRef]

P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
[CrossRef] [PubMed]

der Sande, G. V.

P. Kockaert, P. Tassin, G. V. der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
[CrossRef]

Dolling, G.

Economou, E. N.

T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
[CrossRef]

Egan, P.

A. D. Boardman, P. Egan, R. C. Mitchell-Thomas, Y. G. Rapoport, and N. J. King, “Weakly and strongly nonlinear waves in negative phase metamaterials,” Proc. SPIE 7029, 70291F (2008).
[CrossRef]

Engheta, N.

A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, “Single-negative, double-negative and low index metamaterials and their electromagnetic applications,” IEEE Trans. Antennas Propag. 49, 23-36 (2007).

N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
[CrossRef]

Enkrich, C.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Erentok, A.

A. Alu, N. Engheta, A. Erentok, and R. W. Ziolkowski, “Single-negative, double-negative and low index metamaterials and their electromagnetic applications,” IEEE Trans. Antennas Propag. 49, 23-36 (2007).

Gelens, L.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
[CrossRef]

L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
[CrossRef]

P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
[CrossRef] [PubMed]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, “Crystal optics with spatial dispersion” in Progress in Optics, E.Wolf, ed. (North-Holland, 1971), Vol. IX, Chap. 6, pp. 235-280.
[CrossRef]

Gomila, D.

L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
[CrossRef]

L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
[CrossRef]

Greegor, R. B.

C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
[CrossRef] [PubMed]

Gundogdu, T. F.

T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
[CrossRef]

Guven, M.

Herrero, R.

K. Staliunas, C. Serrat, R. Herrero, C. Cojocaru, and J. Trull, “Subdiffractive light pulses in photonic crystals,” Phys. Rev. E 74, 016605 (2006).
[CrossRef]

Houck, A. A.

A. A. Houck, J. B. Brock, and I. L. Chuang, “Experimental observations of a left-handed material that obeys Snell's law,” Phys. Rev. Lett. 90, 137401 (2003).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1925).

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A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
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P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
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P. Kockaert, P. Tassin, G. V. der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
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C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
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T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
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T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
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T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
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A. D. Boardman, P. Egan, R. C. Mitchell-Thomas, Y. G. Rapoport, and N. J. King, “Weakly and strongly nonlinear waves in negative phase metamaterials,” Proc. SPIE 7029, 70291F (2008).
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D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
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R. A. Shelby, D. R. Smith, S. C. Nemat-Nasser, and S. Schultz, “Microwave transmission through a two-dimensional, isotropic, left-handed metamaterial,” Appl. Phys. Lett. 78, 489-491 (2001).
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Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
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C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental verification and simulation of negative index of refraction using Snell's law,” Phys. Rev. Lett. 90, 107401 (2003).
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T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
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M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics Nanostruct. Fundam. Appl. 6, 87-95 (2008).
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A. D. Boardman, P. Egan, R. C. Mitchell-Thomas, Y. G. Rapoport, and N. J. King, “Weakly and strongly nonlinear waves in negative phase metamaterials,” Proc. SPIE 7029, 70291F (2008).
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M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics Nanostruct. Fundam. Appl. 6, 87-95 (2008).
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P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

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G. D'Aguanno, N. Mattiucci, M. Scalora, and M. J. Bloemer, “Bright and dark gap solitons in a negative index Fabry-Perot etalon,” Phys. Rev. Lett. 93, 213902 (2004).
[CrossRef] [PubMed]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
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R. A. Shelby, D. R. Smith, S. C. 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] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics Nanostruct. Fundam. Appl. 6, 87-95 (2008).
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I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529-534 (2006).
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A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
[CrossRef]

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

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

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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]

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A. D. Boardman, K. Marinov, D. I. Pushkarov, and A. Shivarova, “Influence of nonlinearly induced diffraction on spatial solitary waves,” Opt. Quantum Electron. 32, 49-62 (2000).
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M. Rahm, D. Schurig, D. A. Roberts, S. A. Cummer, D. R. Smith, and J. B. Pendry, “Design of electromagnetic cloaks and concentrators using form-invariant coordinate transformations of Maxwell's equations,” Photonics Nanostruct. Fundam. Appl. 6, 87-95 (2008).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 6, 788-792 (2004).
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R. A. Shelby, D. R. Smith, S. C. 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] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
[CrossRef] [PubMed]

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G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53-55 (2007).
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T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
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K. Staliunas, C. Serrat, R. Herrero, C. Cojocaru, and J. Trull, “Subdiffractive light pulses in photonic crystals,” Phys. Rev. E 74, 016605 (2006).
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P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. 92, 203111 (2008).
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P. Kockaert, P. Tassin, G. V. der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
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L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
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T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
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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).
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L. Gelens, D. Gomila, G. Van der Sande, J. Danckaert, P. Colet, and M. A. Matías, “Dynamical instabilities of dissipative solitons in nonlinear optical cavities with nonlocal materials,” Phys. Rev. A 77, 033841 (2008).
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P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
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L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
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P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. 92, 203111 (2008).
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P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
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L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, “Impact of nonlocal interactions in dissipative systems: towards minimal-sized localized structures,” Phys. Rev. A 75, 063812 (2007).
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P. Kockaert, P. Tassin, G. V. der Sande, I. Veretennicoff, and M. Tlidi, “Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials,” Phys. Rev. A 74, 033822 (2006).
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D. R. Smith, W. J. Padilla, D. C. Vier, D. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184-4187 (2000).
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G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53-55 (2007).
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C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 6, 788-792 (2004).
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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).
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Zharov, A. A.

I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, “Second-harmonic generation in nonlinear left-handed metamaterials,” J. Opt. Soc. Am. B 23, 529-534 (2006).
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A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
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A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
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C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
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C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett. 95, 203901 (2005).
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Appl. Phys. Lett. (4)

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

T. F. Gundogdu, I. Tsiapa, A. Kostopoulos, G. Konstantinidis, N. Katsarakis, R. S. Penciu, M. Kafesaki, E. N. Economou, T. Koschny, and C. M. Soukoulis, “Experimental demonstration of negative magnetic permeability in the far-infrared frequency regime,” Appl. Phys. Lett. 89, 084103 (2006).
[CrossRef]

P. Tassin, X. Sahyoun, and I. Veretennicoff, “Miniaturization of photonic waveguides by the use of left-handed materials,” Appl. Phys. Lett. 92, 203111 (2008).
[CrossRef]

A. A. Zharov, N. A. Zharova, I. V. Shadrivov, and Y. S. Kivshar, “Subwavelength imaging with opaque nonlinear left-handed lenses,” Appl. Phys. Lett. 87, 091104 (2005).
[CrossRef]

Chaos (1)

P. Tassin, L. Gelens, J. Danckaert, I. Veretennicoff, G. Van der Sande, P. Kockaert, and M. Tlidi, “Dissipative structures in left-handed material cavity optics,” Chaos 17, 037116 (2007).
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IEEE Antennas Wireless Propag. Lett. (1)

N. Engheta, “An idea for thin subwavelength cavity resonators using metamaterials with negative permittivity and permeability,” IEEE Antennas Wireless Propag. Lett. 1, 10-13 (2002).
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Figures (1)

Fig. 1
Fig. 1

Cavity containing right-handed (RHM) and left-handed (LHM) materials. The cavity is driven by the input field E i .

Equations (63)

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P [ E ] = ϵ 0 χ ( 1 ) t , r E + P NL [ E ] ,
curl curl E + 1 c 0 2 2 t 2 ( χ ( 1 ) E ) + μ 0 2 t 2 P NL = 0
Δ E 1 c 0 2 χ ( 1 ) 2 E t 2 = grad ( div E ) + μ 0 2 P NL t 2 ,
E ( t , r ) = a ( t , r ) cos ( ω 0 t k 0 r + φ ( t , r ) ) ,
L = 1 2 Δ [ a e + i ( ω 0 t k 0 r + φ ) ] 1 2 χ ( 1 ) c 0 2 2 t 2 [ a e + i ( ω 0 t k 0 r + φ ) ] + c.c. = 1 2 LWE + c.c. ,
LWE = Δ [ A e i ( ω 0 t k 0 r ) ] χ ( 1 ) c 0 2 2 t 2 [ A e i ( ω 0 t k 0 r ) ] .
Δ [ A e i ( ω 0 t k 0 r ) ] = e i ω 0 t Δ [ A e i k 0 r ] ,
Δ [ A e i k 0 r ] = Δ ( e i k 0 r ) A + 2 ( e i k 0 r ) A + e i k 0 r Δ A = Δ ( e i k 0 r ) A 2 ie i k 0 r ( k 0 ) A + e i k 0 r Δ A = [ k 0 2 A 2 i ( k 0 ) A + Δ A ] e i k 0 r .
2 t 2 [ A e i ω 0 t ] = ( ω 0 2 A + 2 i ω 0 A t + 2 A t 2 ) e i ω 0 t .
D ω 0 = 1 c 0 2 ( ω 0 2 + 2 i ω 0 t + 2 t 2 ) ,
= E d t d 3 r .
χ ( 1 ) [ ( D ω 0 A ) e i ( ω 0 t k 0 r ) ] = χ ( 1 ) ( t , r ) ( D ω 0 A ) e i [ ω 0 ( t t ) k 0 ( r r ) ] = e i ( ω 0 t k 0 r ) × χ ( 1 ) ( t , r ) e i ( ω 0 t k 0 r ) ( D ω 0 A ) = e i ( ω 0 t k 0 r ) ( χ ( 1 ) ( t , r ) e i ( ω 0 t k 0 r ) ) ( D ω 0 A ) .
χ 0 ( 1 ) ( t , r ) = χ ( 1 ) ( t , r ) e i ( ω 0 t k 0 r ) ,
e i ( k 0 r ω 0 t ) ( LWE ) = [ k 0 2 A 2 i ( k 0 ) A + Δ A ] χ 0 ( 1 ) ( D ω 0 A ) .
χ 0 ( 1 ) ( D ω 0 A ) = χ 0 ( 1 ) ( t , r ) | ( ω 0 2 c 0 2 + 2 i ω 0 c 0 2 t + 1 c 0 2 2 t 2 ) | t , r A = ( ω 0 2 c 0 2 + 2 i ω 0 c 0 2 t + 1 c 0 2 2 t 2 ) ( χ 0 ( 1 ) A ) = D ω 0 ( χ 0 ( 1 ) A ) .
χ 0 ( 1 ) A = χ 0 ( 1 ) ( t , r ) A ( t t , r r ) .
= 1 c 0 E d τ d 3 r ,
D ω 0 = ρ 0 2 + 2 i ρ 0 τ + 2 τ 2 = ( i ρ 0 + τ ) 2 .
χ 0 ( 1 ) A = X 0 ( τ , r ) U ( τ τ , r r ) = X 0 ( τ , r ) U ( τ , r ) τ X 0 ( τ , r ) | τ | ( τ , r ) U ( r X 0 ( τ , r ) | | ( τ , r ) ) U + O ( 2 ) .
X 0 ( τ , r ) U ( τ , r ) = X 0 U ( τ , r ) ,
τ X 0 ( τ , r ) U τ = τ X 0 U τ ,
( r X 0 ( τ , r ) ) U = x X 0 U x + y X 0 U y + z X 0 U z ,
τ X 0 = τ X e i ( ρ 0 τ k 0 r ) = ( i ρ 0 ) Xe i ( ρ 0 τ k 0 r ) = i ρ 0 X 0 = i c 0 ω 0 χ 0 ( 1 ) .
r X 0 = i k , 0 X 0 = i k , 0 χ 0 ( 1 ) .
X 0 = 1 c 0 F [ X ] 0 = X ̃ 0 c 0 = F [ χ ( 1 ) ] 0 = χ ̃ 0 ( 1 ) = n 2 ( ω 0 , k 0 ) 1 .
X 0 ( τ , r ) U ( τ τ , r r ) = X 0 U τ X 0 U τ ( r X 0 ) U + 1 2 τ 2 X 0 2 U τ 2 + 1 2 x 2 X 0 2 U x 2 + 1 2 y 2 X 0 2 U y 2 + 1 2 z 2 X 0 2 U z 2 + x y X 0 2 U x y + x z X 0 2 U x z + y z X 0 2 U y z + τ x X 0 2 U τ x + τ y X 0 2 U τ y + τ z X 0 2 U τ z + O ( 3 ) .
k 0 = k 0 1 z ,
χ ( 1 ) ( τ , x , y , z ) = χ ( 1 ) ( t , x , y , z ) = χ ( 1 ) ( t , x , y , z ) = χ ( 1 ) ( t , x , y , z ) ,
χ 0 ( 1 ) ( t , x , y , z ) = χ ( 1 ) ( t , x , y , z ) e i ( ω 0 t k 0 z ) = χ 0 ( 1 ) ( t , x , y , z ) = χ 0 ( 1 ) ( t , x , y , z ) = χ 0 ( 1 ) ( t , x , y , z ) e 2 i k 0 z ,
r X 0 = z X 0 1 z ,
x y X 0 = x y X 0 = 0 ,
x τ X 0 = x τ X 0 = 0 .
X 0 ( τ , r ) U ( τ τ , r r ) = X 0 U τ X 0 U τ + 1 2 τ 2 X 0 2 U τ 2 z X 0 U z + 1 2 z 2 X 0 2 U z 2 + 1 2 x 2 X 0 Δ U + O ( 3 ) .
D ω 0 χ 0 ( 1 ) A = A + T + S + M .
A = β 0 2 A ,
T = i β 0 2 ω 0 A t + 1 2 2 β 0 2 ω 0 2 2 A t 2 i c 0 2 ω 0 ( ω 0 n 2 ω 0 ) 3 A t 3 1 2 c 0 2 2 n 2 ω 0 2 4 A t 4 ,
S = i β 0 2 k z A z + 1 2 2 β 0 2 k z 2 2 A z 2 + 1 2 2 β 0 2 k x 2 Δ A ,
M = i 1 ω 0 2 β 0 2 k x 2 Δ A t 1 2 ω 0 2 2 β 0 2 k x 2 Δ 2 A t 2 2 ω 0 β 0 2 k z 2 A t z + i 1 ω 0 2 β 0 2 k z 3 A t 2 z i 1 ω 0 2 β 0 2 k z 2 3 A t z 2 1 2 ω 0 2 2 β 0 2 k z 2 4 A t 2 z 2 .
A = β 0 2 A ,
T = i β 0 2 ω 0 A t + 1 2 2 β 0 2 ω 0 2 2 A t 2 ,
S = i β 0 2 k z A z + 1 2 2 β 0 2 k z 2 2 A z 2 + 1 2 2 β 0 2 k x 2 Δ A ,
M = 2 ω 0 β 0 2 k z 2 A t z .
( LWE ) e i ( ω 0 t k 0 z ) + T + M = ( β 0 2 k 0 2 ) A 2 i k 0 ( 1 β 0 k 0 β 0 k z ) A z + ( 1 2 β 0 2 k z 2 ) 2 A z 2 + ( 1 2 β 0 2 k x 2 ) Δ A .
( LWE ) e i ( ω 0 t k 0 z ) + T + M = 2 i k 0 ( 1 β 0 k 0 β 0 k z ) A z + ( 1 2 β 0 2 k x 2 ) Δ A .
S 4 = 1 4 ! 4 β 2 2 k x 4 Δ 2 A .
A ζ = i [ D + N ] A ,
D = 1 2 k 0 2 β 2 2 2 t 2 ,
N = 3 ω 0 2 c η 0 χ 0 ( 3 ) ( | A | 2 + σ | B | 2 ) A ,
A ( ζ + δ ζ ) = A ( ζ ) + | i δ ζ [ D + N ] | ζ A + O ( 2 ) = e | i δ ζ [ D + N ] | ζ A + O ( 2 ) = e | i δ ζ D | ζ e | i δ ζ N | ζ A + O ( 2 ) = e | i δ ζ N | ζ e | i δ ζ D | ζ A + O ( 2 ) .
D = 1 2 k 0 [ ( 1 1 2 2 β 2 2 k x 2 ) Δ + 1 4 ! 4 β 2 2 k x 4 Δ 2 ] 1 2 k 0 ( 1 + η Δ ) Δ ,
η = 1 4 ! 4 β 2 2 k x 4 .
A ( ζ + δ ζ ) = e i δ ζ [ D + N ] ζ A = 1 + i [ D + N ] ζ + i 2 2 [ D + N ] ζ 2 + O ( 3 ) = 1 + i [ D + N ] ζ 1 2 D 2 δ ζ 2 1 2 N 2 δ ζ 2 1 2 [ D N + N D ] ζ δ ζ 2 + O ( 3 ) .
A ( ζ + δ ζ ) = 1 + i [ D + N ] ζ 1 2 D 2 δ ζ 2 + O ( 3 ) .
A τ = E in ( 1 + i Δ ) A + i δ 2 A + ( i θ 1 2 δ 2 ) 4 A + i Γ | A | 2 A ,
F = 2 π | ρ | 1 | ρ | ,
τ = 2 π F T τ ,
E in = F 2 π E in ,
Δ = F 2 π ψ ,
δ = F 4 π ( l L k L + l R k R ) ,
θ = F 4 π ( l L η L k L + l R η R k R ) ,
Γ = F 2 π ( l L γ L + l R γ R ) ,
θ = F 4 π l L | k L | η L .
θ δ = η L 1 | k L l R k R l L | .

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