Abstract

We analyze linear propagation in negative index materials by starting from a dispersion relation and by deriving the underlying partial differential equation. Transfer functions for propagation are derived in temporal and spatial frequency domains for unidirectional baseband and modulated pulse propagation, as well as for beam propagation. Gaussian beam propagation is analyzed and reconciled with the ray transfer matrix approach as applied to propagation in negative index materials. Nonlinear extensions of the linear partial differential equation are made by incorporating quadratic and cubic terms, and baseband and envelope solitary wave solutions are determined. The conditions for envelope solitary wave solutions are compared with those for the standard nonlinear Schrodinger equation in a positive index material.

© 2006 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. T.-C. Poon and P. P. Banerjee, Contemporary Optical Image Processing with MATLAB (Elsevier, 2001).
  11. P. Tassin, G. Van der Sande, and I. Veretennicoff, "Left-handed materials: the key to subwavelength resolution?" in Proceedings of Symposium IEEE/LEOS Benelux (IEEE, 2004), pp. 41-44.
  12. A. Korpel, "Solitary wave formation through m-th order parametric interaction," Proc. IEEE 67, 1442-1443 (1979).
    [CrossRef]
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  14. V. I. Karpman, Nonlinear Waves in Dispersive Media (Pergamon, 1975).
  15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

2005 (3)

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]

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]

1979 (1)

A. Korpel, "Solitary wave formation through m-th order parametric interaction," Proc. IEEE 67, 1442-1443 (1979).
[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.

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]

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]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

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]

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]

A. Korpel, "Solitary wave formation through m-th order parametric interaction," Proc. IEEE 67, 1442-1443 (1979).
[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, 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]

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]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Savov, S. V.

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

Scalora, M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized nonlinear 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]

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]

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]

Tassin, P.

P. Tassin, G. Van der Sande, and I. Veretennicoff, "Left-handed materials: the key to subwavelength resolution?" in Proceedings of Symposium IEEE/LEOS Benelux (IEEE, 2004), pp. 41-44.

Van der Sande, G.

P. Tassin, G. Van der Sande, and I. Veretennicoff, "Left-handed materials: the key to subwavelength resolution?" in Proceedings of Symposium IEEE/LEOS Benelux (IEEE, 2004), pp. 41-44.

Veretennicoff, I.

P. Tassin, G. Van der Sande, and I. Veretennicoff, "Left-handed materials: the key to subwavelength resolution?" in Proceedings of Symposium IEEE/LEOS Benelux (IEEE, 2004), pp. 41-44.

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]

Zheltikov, A. M.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized nonlinear 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]

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]

Phys. Rev. Lett. (4)

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized nonlinear 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 (2)

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]

A. Korpel, "Solitary wave formation through m-th order parametric interaction," Proc. IEEE 67, 1442-1443 (1979).
[CrossRef]

Rep. Prog. Phys. (1)

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

Other (6)

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

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

P. Tassin, G. Van der Sande, and I. Veretennicoff, "Left-handed materials: the key to subwavelength resolution?" in Proceedings of Symposium IEEE/LEOS Benelux (IEEE, 2004), pp. 41-44.

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

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

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

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Figures (4)

Fig. 1
Fig. 1

Dispersion relation of the form ω = C k , where C = 1 .

Fig. 2
Fig. 2

Propagation of a modulated Gaussian pulse. (a) Initial pulse: ψ ( z = 0 , t ) = f ( t ) = exp ( t τ ) 2 cos ( ω 0 t ) , τ = 0.5 , ω 0 = 20 , (b) Fourier transform of the initial pulse, (c) Fourier transform of the real part of H for z 0 = 2 , C = 200 , (d) initial pulse after propagation at distance z = 2 .

Fig. 3
Fig. 3

Output function ψ ( z , t ) = J 0 ( 2 C z t ) u ( t ) due to a step input.

Fig. 4
Fig. 4

Changes in wavefront radius with propagation distance. Solid line indicates n < 0 ; dotted line n > 0 .

Equations (75)

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ω j t , k j z
ω = W ( k ) , k = K ( ω ) .
ψ ( z , t ) = 1 2 π Ψ ( ω ) exp { j [ ω t K ( ω ) z ] } d ω ,
ψ * ( z , t ) = 1 2 π Ψ * ( ω ) exp { j [ ω t K ( ω ) z ] } d ω = 1 2 π Ψ * ( ω ) exp { j [ ω t + K ( ω ) z ] } d ω ,
ω = W ( k ) = C k , C > 0 ,
ω = ω 0 > 0 , k = k 0 = C ω 0 < 0 ,
ν p ( ω 0 ) = ω k ω 0 = ω 0 2 C < 0 ,
ν g ( ω 0 ) = d ω d k ω 0 = C k 2 ω 0 = ω 0 2 C > 0 .
2 ψ z t + C ψ = 0 .
d Ψ d z = j ( C ω ) Ψ ,
Ψ ( z , ω ) = Ψ ( z = 0 , ω ) exp [ j ( C ω ) z ] = F ( ω ) exp [ j ( C ω ) z ] ,
H ( z , ω ) = Ψ ( z , ω ) Ψ ( z = 0 , ω ) = exp [ j ( C ω ) z ] .
H ( z , ω ) = exp ( j k 0 z ) exp ( j Ω z ν g ) ,
Ψ ( k z , t ) = F z { ψ ( z , t ) } = ψ ( z , t ) exp ( + j k z z ) d z
Ψ ( k z , t ) = Ψ ( k z , t = 0 ) H ( k z , t ) = G ( k z ) exp [ j ( C k z ) t ] .
ψ ( z , t ) = 1 π 0 1 k z sin ( C k z t + k z z ) d k z .
ψ ( z , t ) = J 0 ( 2 C z t ) u ( t ) ,
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 .
ψ ( x , y , z , t ) = Re { ψ e ( x , y , z ) exp [ j ( ω 0 t + k 0 z ) ] } , k 0 = C ω 0 ,
ψ e z = j ω 0 2 C ( 2 ψ e x 2 + 2 ψ e y 2 ) .
Ψ e ( k x , k y , z ) = F x , y { ψ e ( x , y , z ) } = ψ e ( x , y , z ) exp [ j ( k x x + k y y ) ] d x d y .
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 ] ,
H + ( k x , k y , z ) = Ψ e ( k x , k y , z ) Ψ e ( k x , k y , z = 0 ) = exp [ j ( k x 2 + k y 2 ) z 2 k 0 + ] ,
h ( x , y , z ) = j C 2 π ω 0 z exp [ j C 2 ω 0 z ( x 2 + y 2 ) ] .
ψ e = j k 0 + w 0 2 2 q ( z 0 ) exp [ j k 0 + ( x 2 + y 2 ) 2 q ( z 0 ) ] , q ( z 0 ) = z 0 + j k 0 + w 0 2 2 ,
Ψ e ( k x , k y , z = 0 ) = π w 0 2 exp j ( k x 2 + k y 2 ) q ( z 0 ) 2 k 0 + .
z = z f = C z 0 ω 0 k 0 + ,
ψ e ( x , y , z = 0 ) = exp ( x 2 + y 2 ) w 0 2 .
Ψ e ( k x , k y , z ) = π w 0 2 exp j ( k x 2 + k y 2 ) q ω 0 2 C ,
q ( z ) = z j C w 0 2 2 ω 0 .
ψ e ( x , y , z ) = j C w 0 2 2 ω 0 q exp [ j C 2 ω 0 q ( x 2 + y 2 ) ] .
exp [ j C 2 ω 0 q ( x 2 + y 2 ) ] = exp { C 2 w 0 2 2 ω 0 2 2 [ z 2 + ( C w 0 2 2 ω 0 ) 2 ] ( x 2 + y 2 ) } exp { j ( C ω 0 ) z 2 [ z 2 + ( C w 0 2 2 ω 0 ) 2 ] ( x 2 + y 2 ) } .
w 2 ( z ) = 2 z 2 + ( C w 0 2 2 ω 0 ) 2 C 2 w 0 2 2 ω 0 2 , R ( z ) = z 2 + ( C w 0 2 2 ω 0 ) 2 z .
2 ψ z t + C ψ + D ψ 2 = 0 .
ν 2 ψ ξ 2 + C ψ + D ψ 2 = 0 .
2 ν A κ 2 sech 2 ( κ ξ ) 3 sech 2 ( κ ξ ) 2 + C A sech 2 ( κ ξ ) + D A 2 sech 4 ( κ ξ ) = 0 ,
ν = C 4 κ 2 , A = 3 C 2 D .
ψ = 3 C 2 D sech 2 [ κ ( z C 4 κ 2 t ) ] .
2 ψ z t + C ψ + D ψ 3 = 0 .
2 ψ e z t + j ( ω 0 ψ e z + k 0 ψ e t ) + 3 4 D ψ e 2 ψ e * = 0 .
ν 2 ψ e ξ 2 + j ( ω 0 k 0 ν ) ψ e ξ + 3 4 D ψ e 2 ψ e * = 0 .
Re : ν d 2 a d ξ 2 + v a ( d φ d ξ ) 2 ( ω 0 k 0 ν ) a ( d φ d ξ ) + 3 4 D a 3 = 0 ,
Im : ν [ 2 d a d ξ d φ d ξ + a d 2 φ d ξ 2 ] + ( ω 0 k 0 ν ) d a d ξ = 0 .
d φ d ξ = 1 2 ( ω 0 ν C ω 0 ) .
d 2 a d ξ 2 = A 1 a + A 3 a 3 ,
A 1 = 3 4 ( ω 0 ν C ω 0 ) 2 ,
A 3 = 3 4 D ν .
κ = A 1 = 3 2 ( ω 0 ν C ω 0 ) , A = 2 A 1 A 3 = 2 ν D ( ω 0 ν C ω 0 ) 2 = 2 2 ν 3 D κ .
ν = ω 0 2 C + 2 3 κ ω 0 , A = 2 ω 0 κ 3 2 D ( 2 3 κ ω 0 + C ) .
φ = 1 2 ( ω 0 ν C ω 0 ) ( z ω 0 2 κ 3 + C ω 0 t ) = 1 2 ( 2 κ 3 z 2 κ ω 0 3 2 κ 3 + C ω 0 t ) .
ψ = Re { 2 ω 0 κ 3 D 2 ( 2 3 κ ω 0 + C ) sech [ κ ( z ω 0 2 2 3 κ ω 0 + C t ) ] × exp [ j ( ω 0 2 1 + C 3 ( 2 κ ω 0 ) t κ 3 z ) ] exp [ j ( ω 0 t + k 0 z ) ] } .
ν g = ω 0 2 κ 3 + C ω 0 ω 0 2 C ν g ,
ν p = ω 0 [ 1 + 1 2 + C 3 ( ω 0 κ ) ] C ω 0 ( 1 1 C 3 ( ω 0 κ ) ) ω 0 2 C ν p ,
ω = C k 3 D a 2 ( 4 k ) ,
ω ω 0 = ( ω k ) ω 0 ( k + k 0 ) + 1 2 ( 2 ω k 2 ) ω 0 ( k + k 0 ) 2 + ( ω a 2 ) ω 0 a 2 ,
( ω k ) ω 0 = ω 0 2 C , ( 2 ω k 2 ) ω 0 = 2 ω 0 3 C 2 , ( ω a 2 ) ω 0 = 3 D ω 0 4 C .
2 ψ z t + C ψ = 0 .
Ψ ( k z , t = 0 ) = 1 2 [ π δ ( k z ) + j 2 k z ] .
Ψ ( k z , t ) = Ψ ( k z , t = 0 ) exp ( j C k z t ) = 1 2 [ π δ ( k z ) + j 2 k z ] exp ( j C k z t ) .
ψ ( z , t ) = 1 2 π 1 2 [ π δ ( k z ) + j 2 k z ] exp ( j C t k z )
× exp ( j k z z ) d k z = j 2 π 1 k z exp ( j C t k z j k z z ) d k z .
ψ ( z , t ) = j 2 π 1 k z exp ( j C t k z + j k z z ) d k z .
ψ ( z , t ) = 1 π 0 1 k z sin ( C t k z + k z z ) d k z .
for { z > 0 , ψ ( z , t ) = J 0 ( 2 C z t ) , z < 0 , ψ ( z , t ) = 0 .
( x out θ out ) = [ A B C D ] ( x in θ in ) ,
T = ( 1 z 0 1 ) ,
R ̃ = [ 1 0 n + 1 n 1 R 1 n ] ,
R ̃ = [ 1 0 0 1 n ] .
q ( z ) = n q 0 + ( z n z 0 ) .
z = z f = n z 0
z f = d n z 0
R ̃ = [ 1 0 0 n ]

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