Abstract

A solution showing the existence of both bright and dark spatial solitons by a simple modification of a previous approach is obtained for a model of reflection grating with modulated Kerr media without imposing specific restrictions on the physical parameters. The resulted relations of the system parameters and the pertinent physical quantities reveal an interesting algebraic structure and symmetry properties containing the previous results as special cases, with higher degrees of symmetry corresponding to more specific restrictions of the parameters. Detailed analysis of the associated parameter space exhibits clearly delineated regions for the different species of spatial solitons with wide ranging variations of characteristics as functions of the parameters. A practical working scheme is then formulated for tailoring the soliton characteristics according to a relatively simple rule. This is exemplified by a specific choice of soliton characteristics supported by a realistic set of system parameters in connection with the study of the dynamical process of soliton propagation in the system. A further study conducted on the dynamics of linear superpositions of two exact bright soliton pairs shows similar relative-phase dependent, attractive, and repulsive interactions as observed in a homogeneous medium while exhibiting visible envelope broadening due to the diffraction effect. The diffraction effect is shown to result in a wobbling phenomenon for the case of two bright solitons with asymmetric amplitudes. In the case of wide-stripe dark solitons, the propagation features a rapid split-off into several narrow stripes, and the number of new stripes depends proportionally on the width of the initial stripes as well as the ratio between its forward and backward amplitudes. Finally, an enhanced breathing phenomenon was demonstrated as a result of increased nonlinear modulation of the system.

© 2010 Optical Society of America

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  1. Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571-614 (1998).
    [CrossRef]
  2. A. W. Snyder, “Guiding light into the millennium,” IEEE J. Sel. Top. Quantum Electron. 6, 1408-1411 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  12. H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
    [CrossRef]
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    [CrossRef]
  14. H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
    [CrossRef]
  15. H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (1)

2008 (2)

H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
[CrossRef]

C. Rizza, A. Ciattoni, and E. DelRe, “Reflection solitons supported by competing nonlinear gratings,” Phys. Rev. A 78, 013814 (2008).
[CrossRef]

2007 (3)

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity,” Phys. Rev. Lett. 98, 043901 (2007).
[CrossRef] [PubMed]

H. Alatas, “Combined solitons in generalized coupled mode equations of a nonlinear optical Bragg grating,” Phys. Rev. A 76, 023801 (2007).
[CrossRef]

C. Rizza, A. Ciattoni, E. DelRe, and E. Palange, “Counterpropagating reflection grating dark solitons in Kerr media,” Phys. Rev. A 75, 063824 (2007).
[CrossRef]

2006 (2)

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr solitons in reflection gratings,” Opt. Lett. 31, 1507-1509 (2006).
[CrossRef] [PubMed]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
[CrossRef]

2003 (2)

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
[CrossRef]

2002 (4)

2000 (1)

A. W. Snyder, “Guiding light into the millennium,” IEEE J. Sel. Top. Quantum Electron. 6, 1408-1411 (2000).
[CrossRef]

1998 (1)

Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

1996 (1)

D. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015-2032 (1996).
[CrossRef]

1992 (1)

1991 (1)

Afanasjev, V. V.

D. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015-2032 (1996).
[CrossRef]

Alatas, H.

H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
[CrossRef]

H. Alatas, “Combined solitons in generalized coupled mode equations of a nonlinear optical Bragg grating,” Phys. Rev. A 76, 023801 (2007).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
[CrossRef]

Allan, G. R.

Andersen, D. R.

Assanto, G.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

Billeton, T.

Boudebs, G.

Brzdakiewicz, K. A.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

Brzozowski, L.

Cathelinaud, M.

Charpentier, F.

Chauvet, M.

Ciattoni, A.

C. Rizza, A. Ciattoni, and E. DelRe, “Reflection solitons supported by competing nonlinear gratings,” Phys. Rev. A 78, 013814 (2008).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity,” Phys. Rev. Lett. 98, 043901 (2007).
[CrossRef] [PubMed]

C. Rizza, A. Ciattoni, E. DelRe, and E. Palange, “Counterpropagating reflection grating dark solitons in Kerr media,” Phys. Rev. A 75, 063824 (2007).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr solitons in reflection gratings,” Opt. Lett. 31, 1507-1509 (2006).
[CrossRef] [PubMed]

de Luca, A.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

DelRe, E.

C. Rizza, A. Ciattoni, and E. DelRe, “Reflection solitons supported by competing nonlinear gratings,” Phys. Rev. A 78, 013814 (2008).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity,” Phys. Rev. Lett. 98, 043901 (2007).
[CrossRef] [PubMed]

C. Rizza, A. Ciattoni, E. DelRe, and E. Palange, “Counterpropagating reflection grating dark solitons in Kerr media,” Phys. Rev. A 75, 063824 (2007).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr solitons in reflection gratings,” Opt. Lett. 31, 1507-1509 (2006).
[CrossRef] [PubMed]

Fanjoux, G.

Gorza, S.

Hauss, J. W.

Huy, H. P.

Iskandar, A. A.

H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
[CrossRef]

Kandi, A. A.

H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
[CrossRef]

Kinner, S. R.

Kivshar, Y.

Y. Kivshar and G. Stegeman, “Spatial optical solitons: guiding light for future technologies,” Opt. Photonics News 2, 59-63 (2002).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

D. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015-2032 (1996).
[CrossRef]

Luther-Davies, B.

Mel'nikov, I. V.

Nazabal, V.

Palange, E.

C. Rizza, A. Ciattoni, E. DelRe, and E. Palange, “Counterpropagating reflection grating dark solitons in Kerr media,” Phys. Rev. A 75, 063824 (2007).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity,” Phys. Rev. Lett. 98, 043901 (2007).
[CrossRef] [PubMed]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr solitons in reflection gratings,” Opt. Lett. 31, 1507-1509 (2006).
[CrossRef] [PubMed]

Peccianti, M.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

Pelinovsky, D.

Rizza, C.

C. Rizza, A. Ciattoni, and E. DelRe, “Reflection solitons supported by competing nonlinear gratings,” Phys. Rev. A 78, 013814 (2008).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity,” Phys. Rev. Lett. 98, 043901 (2007).
[CrossRef] [PubMed]

C. Rizza, A. Ciattoni, E. DelRe, and E. Palange, “Counterpropagating reflection grating dark solitons in Kerr media,” Phys. Rev. A 75, 063824 (2007).
[CrossRef]

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial Kerr solitons in reflection gratings,” Opt. Lett. 31, 1507-1509 (2006).
[CrossRef] [PubMed]

Sargent, E. H.

Scalora, M.

Sears, J.

Sibilia, C.

Smirl, A. L.

Snyder, A. W.

A. W. Snyder, “Guiding light into the millennium,” IEEE J. Sel. Top. Quantum Electron. 6, 1408-1411 (2000).
[CrossRef]

Soon, B. Y.

Stegeman, G.

Y. Kivshar and G. Stegeman, “Spatial optical solitons: guiding light for future technologies,” Opt. Photonics News 2, 59-63 (2002).
[CrossRef]

Tjia, M. O.

H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
[CrossRef]

Umeton, C.

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

Valkering, T. P.

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
[CrossRef]

Yang, X.

IEEE J. Sel. Top. Quantum Electron. (1)

A. W. Snyder, “Guiding light into the millennium,” IEEE J. Sel. Top. Quantum Electron. 6, 1408-1411 (2000).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (3)

G. Assanto, M. Peccianti, K. A. Brzdakiewicz, A. de Luca, and C. Umeton, “Nonlinear wave propagation and spatial solitons in nematic liquid crystals,” J. Nonlinear Opt. Phys. Mater. 12, 123-134 (2003).
[CrossRef]

H. Alatas, A. A. Kandi, A. A. Iskandar, and M. O. Tjia, “New class of bright spatial solitons obtained by Hirota's method from generalized coupled mode equations of nonlinear optical Bragg grating,” J. Nonlinear Opt. Phys. Mater. 17, 225-233 (2008).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Analytic study of stationary solitons in deep nonlinear Bragg grating,” J. Nonlinear Opt. Phys. Mater. 12, 157-173 (2003).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (4)

Opt. Photonics News (1)

Y. Kivshar and G. Stegeman, “Spatial optical solitons: guiding light for future technologies,” Opt. Photonics News 2, 59-63 (2002).
[CrossRef]

Opt. Quantum Electron. (1)

Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Opt. Quantum Electron. 30, 571-614 (1998).
[CrossRef]

Phys. Rev. A (3)

C. Rizza, A. Ciattoni, E. DelRe, and E. Palange, “Counterpropagating reflection grating dark solitons in Kerr media,” Phys. Rev. A 75, 063824 (2007).
[CrossRef]

C. Rizza, A. Ciattoni, and E. DelRe, “Reflection solitons supported by competing nonlinear gratings,” Phys. Rev. A 78, 013814 (2008).
[CrossRef]

H. Alatas, “Combined solitons in generalized coupled mode equations of a nonlinear optical Bragg grating,” Phys. Rev. A 76, 023801 (2007).
[CrossRef]

Phys. Rev. E (2)

D. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, “Instability-induced dynamics of dark solitons,” Phys. Rev. E 54, 2015-2032 (1996).
[CrossRef]

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E 73, 066606 (2006).
[CrossRef]

Phys. Rev. Lett. (1)

A. Ciattoni, C. Rizza, E. DelRe, and E. Palange, “Counterpropagating spatial solitons in reflection gratings with a longitudinally modulated Kerr nonlinearity,” Phys. Rev. Lett. 98, 043901 (2007).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Bifurcation diagram in the parameter space describing the delineated regions for the admitted solutions of spatial bright (light-gray area) and dark (dark-gray area) solitons for the case of (a) b 0 < 0 and (b) b 0 > 0 . The limiting straight lines related to Γ = 0 are given by 2 b N | b 0 | = ± ( 1 ± b 2 N | b 0 | ) , while the parabolic curves are determined by the relation ( 2 b N | b 0 | ) 2 = 2 ( 1 ± b 2 N | b 0 | ) .

Fig. 2
Fig. 2

Description of ζ ̃ = ( η ̃ η ̃ 0 ) Δ 1 D linear relation with D = 1 for spatial (a) bright ( Δ 1 = 1 ) and (b) dark ( Δ 1 = 2 ) solitons showing the permitted regions as continua of parallel straight lines lying outside the forbidden region bounded by the dash-dot lines with horizontal intercepts η ̃ 0 = ± 1 .

Fig. 3
Fig. 3

Envelope of field magnitude of two bright solitons of the same species and zero relative phase in a system specified in the text for (a) forward and (b) backward fields at z = 0 (dashed curve) and z = 30 (solid curve), with their respective evolutions during the propagation process depicted in (c) and (d).

Fig. 4
Fig. 4

(a) Envelope of field magnitude of two superposed bright solitons of different (+) and (−) species and zero relative phase in a system specified in the text for forward field at z = 0 (dashed curve) and z = 30 (solid curve). (b) The corresponding dynamics of field envelope.

Fig. 5
Fig. 5

(a) Initial forward-field profile of the wide-stripe dark solitons at z = 0 with ( b N , 2 x 0 ) = ( 10 3 , 0.50 ) (solid curve), ( 10 3 , 1.00 ) (dashed curve), and ( 4 × 10 3 , 0.50 ) (dash-dotted curve) in a system specified in the text. Notice the scale differences. (b) The corresponding backward field profiles.

Fig. 6
Fig. 6

Dynamics of (a) forward and (b) backward envelope field magnitude of wide-stripe dark solitons with zero relative phase for the case of ( b N , 2 x 0 ) = ( 10 3 , 0.50 ) in a system specified in the text along with their profile at z = 0 and 30.

Fig. 7
Fig. 7

Dynamics of the (a) forward and (b) backward field profiles of wide-stripe dark solitons for the cases of ( b N , 2 x 0 ) = ( 10 3 , 1.00 ) , with zero relative phase in a system specified in the text.

Fig. 8
Fig. 8

Dynamics of the (a) forward and (b) backward field profiles of wide-stripe dark solitons for the cases of ( b N , 2 x 0 ) = ( 4 × 10 3 , 0.50 ) with zero relative phase in a system specified in the text.

Equations (20)

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

P ̂ f ( b ) U f ( b ) η U f ( b ) + b 0 ( | U f ( b ) | 2 + 2 | U b ( f ) | 2 ) U f ( b ) + c N U b ( f ) + b N ( | U b ( f ) | 2 + 2 | U f ( b ) | 2 ) U b ( f ) + b N U f ( b ) 2 U b ( f ) * + b 2 N U f ( b ) * U b ( f ) 2 = 0.
A D d 2 F d x 2 [ ( η K ) A c N B ] F + [ b 0 A ( A 2 + 2 B 2 ) + b N B ( 3 A 2 + B 2 ) + b 2 N A B 2 ] F 3 = 0 ,
B D d 2 F d x 2 [ ( η + K ) B c N A ] F + [ b 0 B ( B 2 + 2 A 2 ) + b N A ( 3 B 2 + A 2 ) + b 2 N B A 2 ] F 3 = 0.
A ( η K ) + Δ 1 A D ζ + B c N = 0 ,
B ( η + K ) + Δ 1 B D ζ + A c N = 0 ,
2 Δ 2 A D ζ + A b 0 ( A 2 + 2 B 2 ) + B b N ( 3 A 2 + B 2 ) + A B 2 b 2 N = 0 ,
2 Δ 2 B D ζ + B b 0 ( B 2 + 2 A 2 ) + A b N ( 3 B 2 + A 2 ) + B A 2 b 2 N = 0 ,
K ± = ± c N Γ 2 b N ,
B ± = Δ 2 D ζ ( b 0 + b 2 N ± Γ ) 2 b N 2 b 0 ( b 0 + b 2 N ) ,
A ± = 2 b N B ± b 0 + b 2 N ± Γ ,
ζ = c N ( b 0 + b 2 N ) + 2 η b N 2 Δ 1 D b N ,
Γ = ( b 0 + b 2 N ) 2 4 b N 2 .
2 | b N b 0 | | ( 1 b 2 N | b 0 | ) | ,
2 | b N b 0 | | ( 1 + b 2 N | b 0 | ) |
( 2 b N | b 0 | ) 2 > 2 ( 1 b 2 N | b 0 | ) ,
( 2 b N | b 0 | ) 2 < 2 ( 1 b 2 N | b 0 | ) ,
v = ( 2 b N b 0 ) ( 1 + b 2 N b 0 ) ,
ζ ̃ = ( η ̃ η ̃ 0 ) Δ 1 D ,
r ± = | A ± B ± | = | v 1 ± sign ( b 0 + b 2 N ) 1 v 2 |
U f ( b ) 0 ( x , 0 ) = ρ [ U f ( b ) ( x x 0 , 0 ) + U f ( b ) ( x + x 0 , 0 ) e i φ ] ,

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