Abstract

We demonstrate that a class of simplified complex dispersion relations, which obey causality, can model baseband and envelope propagation in conventional and negative index materials. One such dispersion relation is a special case of the Drude model, another yields the Kuramoto–Shivashinsky equation, while a third, in the limit, yields the simplest dispersion for negative index materials.

© 2008 Optical Society of America

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References

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  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]
  2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  3. D. G. Stavenga, “Invertibrate superposition eyes-structures that behave like metamaterial with negative refractive index,” J. Eur. Opt. Soc. Rapid Publ. 1, 06010 (2006).
    [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. P. P. Banerjee and G. Nehmetallah, “Linear and nonlinear propagation in negative index materials,” J. Opt. Soc. Am. B 23, 2348-2355 (2006).
    [CrossRef]
  6. P. P. Banerjee and G. Nehmetallah, “Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials,” J. Opt. Soc. Am. B 24, A69-A76 (2007).
    [CrossRef]
  7. K. E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).
  8. J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
    [CrossRef]
  9. A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC, 2004).
  10. A. D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC, 1999).
  11. P. P. Banerjee, G. Nehmetallah, R. Aylo, and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698, 6698M0-8 (2007).
  12. J. F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting (McGraw-Hill, 1976).
  13. V. I. Karpman, Non-Linear Waves in Dispersive Media, International Series of Monographs in Natural Philosophy, 1st ed., (Pergamon Press, 1974).

2007 (2)

P. P. Banerjee, G. Nehmetallah, R. Aylo, and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698, 6698M0-8 (2007).

P. P. Banerjee and G. Nehmetallah, “Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials,” J. Opt. Soc. Am. B 24, A69-A76 (2007).
[CrossRef]

2006 (2)

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

D. G. Stavenga, “Invertibrate superposition eyes-structures that behave like metamaterial with negative refractive index,” J. Eur. Opt. Soc. Rapid Publ. 1, 06010 (2006).
[CrossRef]

2000 (2)

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]

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

1999 (1)

J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
[CrossRef]

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]

Asakura, T.

K. E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

Aylo, R.

P. P. Banerjee, G. Nehmetallah, R. Aylo, and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698, 6698M0-8 (2007).

Banerjee, P. P.

P. P. Banerjee, G. Nehmetallah, R. Aylo, and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698, 6698M0-8 (2007).

P. P. Banerjee and G. Nehmetallah, “Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials,” J. Opt. Soc. Am. B 24, A69-A76 (2007).
[CrossRef]

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

Buranasiri, P.

P. P. Banerjee, G. Nehmetallah, R. Aylo, and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698, 6698M0-8 (2007).

Claerbout, J. F.

J. F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting (McGraw-Hill, 1976).

Drotar, J. T.

J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
[CrossRef]

Karpman, V. I.

V. I. Karpman, Non-Linear Waves in Dispersive Media, International Series of Monographs in Natural Philosophy, 1st ed., (Pergamon Press, 1974).

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]

Lu, T.-M.

J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
[CrossRef]

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]

Peiponen, K. E.

K. E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

Pendry, J. B.

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

Polyanin, A. D.

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC, 2004).

Poularikas, A. D.

A. D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC, 1999).

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.

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]

Stavenga, D. G.

D. G. Stavenga, “Invertibrate superposition eyes-structures that behave like metamaterial with negative refractive index,” J. Eur. Opt. Soc. Rapid Publ. 1, 06010 (2006).
[CrossRef]

Vartiainen, E. M.

K. E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

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]

Wang, G.-C.

J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
[CrossRef]

Zaitsev, V. F.

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC, 2004).

Zhao, Y.-P.

J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
[CrossRef]

J. Eur. Opt. Soc. Rapid Publ. (1)

D. G. Stavenga, “Invertibrate superposition eyes-structures that behave like metamaterial with negative refractive index,” J. Eur. Opt. Soc. Rapid Publ. 1, 06010 (2006).
[CrossRef]

J. Opt. Soc. Am. B (2)

Phys. Rev. E (1)

J. T. Drotar, Y.-P. Zhao, T.-M. Lu, and G.-C. Wang, “Numerical analysis of the noisy Kuramoto-Sivashinsky equation in 2+1 dimensions,” Phys. Rev. E 59, 177-185 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

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]

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

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]

Proc. SPIE (1)

P. P. Banerjee, G. Nehmetallah, R. Aylo, and P. Buranasiri, “Dispersion relations for negative index materials and slow light,” Proc. SPIE 6698, 6698M0-8 (2007).

Other (5)

J. F. Claerbout, Fundamentals of Geophysical Data Processing with Applications to Petroleum Prospecting (McGraw-Hill, 1976).

V. I. Karpman, Non-Linear Waves in Dispersive Media, International Series of Monographs in Natural Philosophy, 1st ed., (Pergamon Press, 1974).

A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations for Engineers and Scientists (Chapman & Hall/CRC, 2004).

A. D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC, 1999).

K. E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

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Figures (5)

Fig. 1
Fig. 1

(a) Real and (b) imaginary parts of the complex susceptibility using the Drude model for various values of Γ and assuming ω 0 = ω 0 D r = 10 14 rad s . Note that for Γ = 2 ω 0 , the Drude model coincides with our model (6) without the 1 in the real part (top, bold). For comparison, the real part in our model (with the 1 ) is also shown in the top graph (bottom, bold).

Fig. 2
Fig. 2

(a) Real and (b) imaginary parts of the complex propagation constant using the Drude model for various values of Γ and assuming ω 0 = ω 0 D r = 10 14 rad s . For comparison, our model given by Eq. (5) is also superposed. Our model [Eq. (5)] shows the real part of the propagation constant tending to zero, and the imaginary part tending to a nonzero value for large frequencies. Note that our model and the Drude model approximately agree over ω < 0.5 × 10 14 rad s . Without the 1 term in the real part in [Eq. (6)], the Drude model and ours are obviously identical for Γ = 2 ω 0 .

Fig. 3
Fig. 3

Real parts of normalized dispersion relation k ± ( ω ) , k D ( ω ) . Note that around the marked point, the dispersion relation k ( ω ) exhibits negative phase velocity and positive group velocity, characteristic of a NIM.

Fig. 4
Fig. 4

(a) Baseband initial Gaussian pulse (initial width τ = 5 ) in time domain and after propagating a distance Z = 10 . (b) Energy decay of the propagating pulse.

Fig. 5
Fig. 5

Gaussian pulse envelope in time domain in a NIM for ω c n = 2 and initial width τ = 20 . (a) Initial pulse, (b) after propagation by Z = 10 .

Equations (21)

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χ r D r ( ω ) ω 0 D r 2 ω 2 ( ω 0 D r 2 ω 2 ) 2 + Γ 2 ω 2 ,
χ i D r ( ω ) Γ ω ( ω 0 D r 2 ω 2 ) 2 + Γ 2 ω 2 ,
χ r ( ω ) = ( k r 2 k i 2 ) ( c ω ) 2 1 , χ i ( ω ) = ( 2 k r k i ) ( c ω ) 2 ,
k r = sgn ( ω ) { ω 2 2 c 2 [ ( χ r + 1 ) + ( χ r + 1 ) 2 + χ i 2 ] } 1 2 ,
k i = { ω 2 2 c 2 [ ( χ r + 1 ) + ( χ r + 1 ) 2 + χ i 2 ] } 1 2 .
k ± ( ω ) = k r ± ( ω ) j k i ± ( ω ) ± ( ω 0 ν ) [ ( ω ω 0 ) 1 + ( ω ω 0 ) 2 + j 1 1 + ( ω ω 0 ) 2 ] ,
k D ( ω ) = k + ( ω ) j 1 k r D ( ω ) j k i D ( ω ) ( ω 0 ν ) [ ( ω ω 0 ) 1 + ( ω ω 0 ) 2 j ( ω ω 0 ) 2 1 + ( ω ω 0 ) 2 ] ,
χ D ( ω ) = χ r D ( ω ) j χ i D ( ω ) ( c ν ) 2 { 1 ( ω ω 0 ) 2 [ 1 + ( ω ω 0 ) 2 ] 2 1 } j { 2 ( ω ω 0 ) [ 1 + ( ω ω 0 ) 2 ] 2 } .
B ( ω ; z ) P ( ω ; z ) = exp [ j k ( ω ) z ] = exp { j [ k r ( ω ) j k i ( ω ) ] z } ,
ln B j ϕ B j ϕ P = k i z j k r z .
ln B = k i z , ϕ P = k r z ϕ B .
ψ z 1 ω 0 2 3 ψ z t 2 + 1 ν ψ t 1 ν ω 0 2 ψ t 2 = 0 .
ψ n Z 3 ψ n Z T 2 + ψ n T 2 ψ n T 2 = 0 , ψ n ( Z , T ) = ψ ( z , t ) .
k n D ( ω n ) ω n ω n 3 j ( ω n 2 ω n 4 ) .
ψ n Z + ψ n T + 3 ψ n T 3 2 ψ n T 2 4 ψ n T 4 = 0 .
ψ n ( Z , T ) = I t 1 { Ψ n ( Z , ω n ) } = I t 1 { Ψ n ( 0 , ω n ) exp [ ω n 2 j ω n 1 + ω n 2 Z ] } ,
ψ n Z 3 ψ n Z T 2 ψ n T + ψ n = 0 .
ω c n = ω c ω 0 , k c n = k c ω 0 ν = ω c n 1 + ω c n 2 ,
a ψ e n Z + b 3 ψ e n Z T 2 + j c 2 ψ e n T Z + j d 2 ψ e n T 2 + e ψ e n T + f ψ e n = 0 ,
a = ( 1 + ω c n 2 ) , b = 1 , c = 2 ω c n , d = k c n , e = 2 k c n ω c n + 1 , f = 1 .
ψ e n ( Z , T ) = I t 1 { Ψ e n ( Z , ω n ) } = I t 1 { Ψ e n ( 0 , ω n ) exp [ ( d ω n 2 j e ω n f b ω n 2 j c ω n a ) Z ] } .

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