Abstract

Starting from a simple dispersion relation for negative index materials and a heuristic nonlinear Klein–Gordon-type extension, we derive the evolution equations for the envelopes of beams and spatiotemporal pulses in nonlinear dispersive negative index media. Using existing numerical methods, based on fast Fourier–Bessel transforms, we study the stability of the solitary wave solutions. Nonlinearity and dispersion management are then incorporated to find stable solutions of the underlying partial differential equations.

© 2007 Optical Society of America

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  1. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  2. J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
    [CrossRef]
  3. S. A. Ramakrishna, "Physics of negative refractive index materials," Rep. Prog. Phys. 68, 449-521 (2005).
    [CrossRef]
  4. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000).
    [CrossRef] [PubMed]
  5. C. G. Parazzoli, R. B. Greegor, K. Li, B. E. C. Kontenbah, and M. H. Tanielian, "Experimental verification and simulation of negative index of refraction using Snell's law," Phys. Rev. Lett. 90, 107401 (2003).
    [CrossRef] [PubMed]
  6. M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
    [CrossRef]
  7. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized nonlinear Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials," Phys. Rev. Lett. 95, 013902 (2005).
    [CrossRef] [PubMed]
  8. A. Korpel and P. P. Banerjee, "A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions," Proc. IEEE 72, 1109-1130 (1984).
    [CrossRef]
  9. K. E. Lonngren and S. V. Savov, Fundamentals of Electromagnetics with MATLAB (Scitech, 2005).
  10. P. P. Banerjee and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B 23, 2348-2355 (2006).
    [CrossRef]
  11. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  12. T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).
  13. P. P. Banerjee, G. Nehmetallah, and M. Chatterjee, "Numerical modeling of cylindrically symmetric nonlinear self-focusing using an adaptive fast Hankel split-step method," Opt. Commun. 249, 293-300 (2005).
    [CrossRef]
  14. G. Nehmetallah and P. P. Banerjee, "Numerical modeling of (D+1)-dimensional solitons in a sign alternating nonlinear medium using an adaptive fast Hankel split step method," J. Opt. Soc. Am. B 22, 2200-2207 (2005).
    [CrossRef]
  15. G. Nehmetallah and P. P. Banerjee, "Stabilization of a (D+1)-dimensional dispersion managed solitons in Kerr media by an alternating dispersion structure," J. Opt. Soc. Am. B 23, 203-211 (2006).
    [CrossRef]
  16. M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
    [CrossRef]
  17. J. Kaestel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, "Negative refraction without absorption in the optical regime," Presented at the 37th Meeting of the Division of Atomic, Molecular and Optical Physics, Knoxville, Tenn., May 16-20, 2006.
  18. J. Woodley and M. Mojahedi, "Backward wave propagation in left-handed media with isotropic and anisotropic permittivity tensors," J. Opt. Soc. Am. B 23, 2377-2382 (2006).
    [CrossRef]
  19. V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, 1975).
  20. 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]

2007 (1)

M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

2006 (4)

2005 (5)

G. Nehmetallah and P. P. Banerjee, "Numerical modeling of (D+1)-dimensional solitons in a sign alternating nonlinear medium using an adaptive fast Hankel split step method," J. Opt. Soc. Am. B 22, 2200-2207 (2005).
[CrossRef]

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

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

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

P. P. Banerjee, G. Nehmetallah, and M. Chatterjee, "Numerical modeling of cylindrically symmetric nonlinear self-focusing using an adaptive fast Hankel split-step method," Opt. Commun. 249, 293-300 (2005).
[CrossRef]

2004 (1)

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

2003 (1)

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

2000 (2)

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

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

1984 (1)

A. Korpel and P. P. Banerjee, "A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions," Proc. IEEE 72, 1109-1130 (1984).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Banerjee, P. P.

P. P. Banerjee and G. Nehmetallah, "Linear and nonlinear propagation in negative index materials," J. Opt. Soc. Am. B 23, 2348-2355 (2006).
[CrossRef]

G. Nehmetallah and P. P. Banerjee, "Stabilization of a (D+1)-dimensional dispersion managed solitons in Kerr media by an alternating dispersion structure," J. Opt. Soc. Am. B 23, 203-211 (2006).
[CrossRef]

P. P. Banerjee, G. Nehmetallah, and M. Chatterjee, "Numerical modeling of cylindrically symmetric nonlinear self-focusing using an adaptive fast Hankel split-step method," Opt. Commun. 249, 293-300 (2005).
[CrossRef]

G. Nehmetallah and P. P. Banerjee, "Numerical modeling of (D+1)-dimensional solitons in a sign alternating nonlinear medium using an adaptive fast Hankel split step method," J. Opt. Soc. Am. B 22, 2200-2207 (2005).
[CrossRef]

A. Korpel and P. P. Banerjee, "A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions," Proc. IEEE 72, 1109-1130 (1984).
[CrossRef]

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Chatterjee, M.

P. P. Banerjee, G. Nehmetallah, and M. Chatterjee, "Numerical modeling of cylindrically symmetric nonlinear self-focusing using an adaptive fast Hankel split-step method," Opt. Commun. 249, 293-300 (2005).
[CrossRef]

D'Aguanno, G.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

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

Fan, D.

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]

Fleischhauer, M.

J. Kaestel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, "Negative refraction without absorption in the optical regime," Presented at the 37th Meeting of the Division of Atomic, Molecular and Optical Physics, Knoxville, Tenn., May 16-20, 2006.

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]

Greegor, R. B.

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

Haus, J. W.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

Kaestel, J.

J. Kaestel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, "Negative refraction without absorption in the optical regime," Presented at the 37th Meeting of the Division of Atomic, Molecular and Optical Physics, Knoxville, Tenn., May 16-20, 2006.

Karpman, V. I.

V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, 1975).

Kontenbah, B. E. C.

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

Korpel, A.

A. Korpel and P. P. Banerjee, "A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions," Proc. IEEE 72, 1109-1130 (1984).
[CrossRef]

Li, K.

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

Lonngren, K. E.

K. E. Lonngren and S. V. Savov, Fundamentals of Electromagnetics with MATLAB (Scitech, 2005).

Mattiucci, N.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

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

Mojahedi, M.

Nehmetallah, G.

Nemat-Nasser, S. C.

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

Padilla, W. J.

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

Parazzoli, C. G.

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

Pendry, J. B.

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Poon, T.-C.

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

Ramakrishna, S. A.

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

Savov, S. V.

K. E. Lonngren and S. V. Savov, Fundamentals of Electromagnetics with MATLAB (Scitech, 2005).

Scalora, M.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

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

Schultz, S.

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

Smith, D. R.

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

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

Stockman, M. I.

M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

Su, W.

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 Schrodinger equation for dispersive susceptibility and permeability: application to negative index materials," Phys. Rev. Lett. 95, 013902 (2005).
[CrossRef] [PubMed]

Tanielian, M. H.

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

Vier, D. C.

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

Walsworth, R. L.

J. Kaestel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, "Negative refraction without absorption in the optical regime," Presented at the 37th Meeting of the Division of Atomic, Molecular and Optical Physics, Knoxville, Tenn., May 16-20, 2006.

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]

Wen, S.

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]

Woodley, J.

Xiang, 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]

Yelin, S. F.

J. Kaestel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, "Negative refraction without absorption in the optical regime," Presented at the 37th Meeting of the Division of Atomic, Molecular and Optical Physics, Knoxville, Tenn., May 16-20, 2006.

Zheltikov, A. M.

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

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

Appl. Phys. B (1)

M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, J. W. Haus, and A. M. Zheltikov, "Negative refraction of ultra-short pulses," Appl. Phys. B 81, 393-402 (2005).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (1)

P. P. Banerjee, G. Nehmetallah, and M. Chatterjee, "Numerical modeling of cylindrically symmetric nonlinear self-focusing using an adaptive fast Hankel split-step method," Opt. Commun. 249, 293-300 (2005).
[CrossRef]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (5)

M. I. Stockman, "Criterion for negative refraction with low optical losses from a fundamental principle of causality," Phys. Rev. Lett. 98, 177404 (2007).
[CrossRef]

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

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

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

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

Phys. Today (1)

J. B. Pendry and D. R. Smith, "Reversing light with negative refraction," Phys. Today 57, 37-43 (2004).
[CrossRef]

Proc. IEEE (1)

A. Korpel and P. P. Banerjee, "A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions," Proc. IEEE 72, 1109-1130 (1984).
[CrossRef]

Rep. Prog. Phys. (1)

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

Other (5)

K. E. Lonngren and S. V. Savov, Fundamentals of Electromagnetics with MATLAB (Scitech, 2005).

J. Kaestel, M. Fleischhauer, S. F. Yelin, and R. L. Walsworth, "Negative refraction without absorption in the optical regime," Presented at the 37th Meeting of the Division of Atomic, Molecular and Optical Physics, Knoxville, Tenn., May 16-20, 2006.

V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, 1975).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).

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Figures (10)

Fig. 1
Fig. 1

FFTSS method for 1-D pulse propagation in NIM.

Fig. 2
Fig. 2

The initial pulse (fine dots), the analytical solution (bold curve), and the numerical solution (coarse dots) using the FFTSS method [11] after propagating for some time t = 4 with D = 1 , κ = 1 , C = 200 , ω 0 = 20 .

Fig. 3
Fig. 3

FFTSS method for 3-D pulse propagation in NIM.

Fig. 4
Fig. 4

Townes soliton profiles.

Fig. 5
Fig. 5

Defocusing of a ( 2 + 1 + 1 ) -dimensional spatiotemporal optical pulses in NIM after propagating a time τ = 1.2 . (a) 3-D view, (b) top view.

Fig. 6
Fig. 6

Focusing of a ( 2 + 1 + 1 ) -dimensional optical pulses in NIM after propagating a time τ = 1.2 . (a) 3-D view, (b) top view.

Fig. 7
Fig. 7

Spatial evolution of stable ( 2 + 0 + 1 ) -dimensional solitary waves through nonlinearity management when A 0 = 2 5 , w 0 = 5 , D 0 = 1 , D 1 = 0.5 , and period = 0.02 , which satisfy the stability conditions derived in [14].

Fig. 8
Fig. 8

Spatiotemporal evolution of stable ( 2 + 1 + 1 ) -dimensional solitary waves through nonlinearity management when A 0 = 0.1 , w 0 = 10 , D 0 = 1 , D 1 = 0.5 , and period = 0.4 , which satisfy the stability conditions derived in [14].

Fig. 9
Fig. 9

Spatial evolution of stable ( 2 + 0 + 1 ) -dimensional solitary waves through dispersion/diffraction management, with the parameters C 0 = 1 , C 1 = 0.1 , A 0 = 1.5 , w 0 = 1 , and period = 0.04 , which satisfy the stability conditions derived in [15].

Fig. 10
Fig. 10

Spatiotemporal evolution of stable ( 2 + 1 + 1 ) -dimensional solitary wave generation through dispersion/diffraction management when C 0 = 1 , C 1 = 0.1 , A 0 = 0.2 , w 0 = 10 , and period = 3 , which satisfy the stability conditions derived in [15].

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

ω = W ( k ) = C k , C > 0 ,
ω = ω 0 > 0 , k = k 0 = C ω 0 < 0 ,
v g ( ω 0 ) = d ω d k ω 0 = C k 2 ω 0 = ω 0 2 C > 0 ,
v p ( ω 0 ) = ω k ω 0 = ω 0 2 C < 0 .
ω j t , k j z
2 ψ z t + C ψ = 0 .
2 ψ z t + C ψ + α ψ z D ̃ 3 ψ z 3 = 0 .
k = k x 2 + k y 2 + k z 2 k z + 1 2 ( k x 2 + k y 2 ) k z 1 C ω 1 ,
ω [ k z 2 + 1 2 ( k x 2 + k y 2 ) ] C k z .
ω j t , k x j x , k y j y , k z j z
3 ψ z 2 t + 1 2 3 ψ x 2 t + 1 2 3 ψ y 2 t + C ψ z = 0 .
3 ψ z 2 t + 1 2 3 ψ x 2 t + 1 2 3 ψ y 2 t + C ψ z + α [ 2 ψ z 2 + 1 2 ( 2 ψ x 2 + 2 ψ y 2 ) ] = 0 .
2 ψ z t + C ψ + D ψ 3 = 0 .
3 ψ z 2 t + C ψ z + D ψ 3 z 3 = 0 .
3 ψ z 2 t + 1 2 3 ψ x 2 t + 1 2 3 ψ y 2 t + C ψ z + D ψ 3 z = 0 .
ψ ( x , y , z , t ) = Re { ψ e ( x , y , z , t ) exp [ j ( ω 0 t + k 0 z ) ] } , k 0 = C ω 0
2 j C ω 0 2 ψ e z t + j ω 0 2 ( 2 ψ e x 2 + 2 ψ e y 2 + 2 2 ψ e z 2 ) C 2 ω 0 2 ψ e t C ψ e z + j 3 D C 4 ω 0 ψ e 2 ψ e + 1 2 ( 3 ψ e x 2 t + 3 ψ e y 2 t + 2 3 ψ e z 2 t ) = 0 .
2 j C ω 0 2 ψ e z t + j ω 0 2 ( 2 ψ e x 2 + 2 ψ e y 2 + 2 2 ψ e z 2 ) C 2 ω 0 2 ( ψ e t + ω 0 2 C ψ e z ) + j 3 D C 4 ω 0 ψ e 2 ψ e = 0 .
2 j k 0 2 ψ e z t + ( α + j ω 0 2 ) ( 2 ψ e x 2 + 2 ψ e y 2 + 2 2 ψ e z 2 ) ( C ψ e z + k 0 2 ψ e t ) + j 3 D k 0 4 ψ e 2 ψ e = 0 ,
2 ψ e z t + ω 0 2 2 C 2 ψ e z 2 + j 2 ( ω 0 ψ e z + k 0 ψ e t ) + 3 8 D ψ e 2 ψ e * = 0 .
( v ω 0 2 2 C ) 2 ψ e ξ 2 + j 2 ( ω 0 k 0 v ) ψ e ξ + 3 8 D ψ e 2 ψ e * = 0 .
d 2 a d ξ 2 = A 1 a + A 3 a 3 ,
A 1 = 3 ( ω 0 C v ω 0 ) 2 16 ( v ω 0 2 2 C ) , A 3 = 3 D 8 ( v ω 0 2 2 C ) ,
a = A sech ( κ ξ ) , A = 2 A 1 A 3 , κ = A 1 .
ψ e z = j ω 0 2 C ( 2 ψ e x 2 + 2 ψ e y 2 ) + j 3 D 4 ω 0 ψ e 2 ψ e .
2 a = 2 k 0 κ ̃ a 3 D 2 C ( k 0 ) 2 a 3 ,
2 a = 2 k 0 + κ ̃ + a 2 n 2 ( k 0 + ) 2 a 3 .
a = A sech ( K x ) ,
A = 2 κ ̃ + n 2 k 0 + ,
K = 2 κ ̃ + k 0 + .
a = ( κ ̃ + n 2 k 0 + ) a ̃ , r = ( 2 κ ̃ + k 0 + ) 1 2 r ̃ ,
T 2 = d 2 d r 2 + 1 r d d r ,
d 2 a ̃ d r ̃ 2 + 1 r ̃ d a ̃ d r ̃ a ̃ + a ̃ 3 = 0 ,
a = A sech ( K x ) ,
A = 8 κ ̃ 3 D k 0 ,
K = 2 κ ̃ k 0 .
ψ e = 8 C κ ̃ 3 D k 0 sech ( 2 κ ̃ k 0 x ) exp j κ ̃ z ,
ψ e = 2 κ ̃ + n 2 k 0 + sech ( 2 κ ̃ + k 0 + x ) exp j κ ̃ + z
a = ( 4 C κ ̃ 3 D k 0 ) a ̃ , r = ( 2 κ ̃ k 0 ) 1 2 r ̃ ,
ξ = z ω 0 2 C t , τ = t ,
j C 2 ω 0 ψ e τ + 2 ψ e ξ τ + ω 0 2 4 C ( 2 ψ e x 2 + 2 ψ e y 2 2 2 ψ e ξ 2 ) + 3 D 8 ψ e 2 ψ e = 0 .
2 ψ e z t + ω 0 2 4 C ( z ) ( 2 ψ e x 2 + 2 ψ e y 2 + 2 2 ψ e z 2 ) ω 0 2 j ( C ( z ) ω 0 2 ψ e t + ψ e z ) + 3 D ( z ) 8 ψ e 2 ψ e = 0 .
H ( k x , k y , z ) = Ψ e ( k x , k y , z ) Ψ e ( k x , k y , z = 0 ) = exp [ j ( ω 0 2 C ) ( k x 2 + k y 2 ) z ] ,
ψ e z = j ω 0 2 C ( z ) ( 2 ψ e x 2 + 2 ψ e y 2 ) + j 3 D ( z ) 4 ω 0 ψ e 2 ψ e .
z ̃ = z L d , x ̃ = x w 0 , y ̃ = y w 0 , u = ( L d ) 1 2 ψ e ,
L d = C 0 ω 0 w 0 2 , D ̃ ( z ) = 3 D ( z ) 4 D 0 ω 0 , C ̃ ( z ) = C ( z ) C 0 ,
j u z + 1 2 C ( z ) ( 2 u x 2 + 2 u y 2 ) + D ( z ) u 2 u = 0 ,
j u z + 1 2 C ( z ) 1 r r ( r u r ) + D ( z ) u 2 u = 0 .
z ̃ = C 0 ω 0 z , x ̃ = C 0 ω 0 x , y ̃ = C 0 ω 0 y , u = ( 3 D 0 4 C 0 ) 1 2 ψ e ,
τ = ( C 0 k 2 ω 0 ) 1 2 ( t C 0 ω 0 2 z ) , D ̃ ( z ) = D ( z ) D 0 ,
C ̃ ( z ) = C ( z ) C 0 , k 2 = 2 C 0 ω 0 3
j u z + 1 2 C ( z ) ( 2 u x 2 + 2 u y 2 sgn ( k 2 ) 2 u τ 2 ) + 2 u z 2 + D ( z ) u 2 u = 0 ,
j u z + 1 2 C ( z ) 1 r 2 r ( r 2 u r ) + D ( z ) u 2 u = 0 .
j u z + 1 2 1 r d ̃ 1 r ( r d ̃ 1 u r ) + D ( z ) u 2 u = 0 ,
u = u ( r , z ) = A ( z ) exp [ r 2 2 w 2 ( z ) + j b ( z ) r 2 ] .
d 2 w d z 2 = γ d ̃ 1 α d ̃ + 1 1 w 3 d ̃ N 0 β d ̃ 1 2 α d ̃ + 1 D ( z ) w d ̃ + 1 ,
α d ̃ = 0 r d ̃ exp ( r 2 ) d r , β d ̃ = 0 r d ̃ exp ( 2 r 2 ) d r ,
γ d ̃ 1 = α d ̃ + 1 ,
γ 1 N 0 β 1 > 1 , D 1 < 6 ( γ 1 N 0 β 1 1 ) ,
j u z + 1 2 C ( z ) 1 r d ̃ r ( r d ̃ u r ) + u 2 u = 0 ,
d 2 w d z 2 = γ d ̃ 1 α d ̃ + 1 C 2 ( z ) w 3 N 0 d ̃ β d ̃ 1 2 α d ̃ + 1 C ( z ) w d ̃ + 1 + 1 C ( z ) d w d z d C d z ,

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