Abstract

We study the formation of dark-bright vector soliton pairs in nonlocal Kerr-type nonlinear medium. We show, by analytical analysis and direct numerical calculation, that in addition to stabilize of vector soliton pairs nonlocal nonlinearity also helps to reduce the threshold power for forming a guided bright soliton. With help of the nonlocality, it is expected that the observation of dark-bright vector soliton pairs in experiments becomes more workable.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. W. Synder and D. J. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
    [CrossRef]
  2. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
    [CrossRef]
  3. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
    [CrossRef] [PubMed]
  4. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
    [CrossRef]
  5. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
    [CrossRef]
  6. Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171-3173 (2005).
    [CrossRef] [PubMed]
  7. G. C. Duree, et al., "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
    [CrossRef] [PubMed]
  8. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  9. C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
    [CrossRef] [PubMed]
  10. A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
    [CrossRef] [PubMed]
  11. D. N. Christodoulides and R. I. Joseph, "Vector solitons in birefringent nonlinear dispersive media," Opt. Lett. 13, 53-55 (1988).
    [CrossRef] [PubMed]
  12. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, 2003).
  13. A. P. Sheppard and Yu. S. Kivshar, "Polarized dark solitons in isotropic Kerr media," Phys. Rev. E 55, 4773-4782 (1997).
    [CrossRef]
  14. M. Shalaby and A. J. Marthelemy, "Observation of the self-guided propagation of a dark and bright spatial solitonpair in a focusing nonlinear medium," IEEE J. Quantum Electron. 28, 2736-2741 (1992).
    [CrossRef]
  15. Z. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, Yu. S. Kivshar, and V. V. Afanasjev, "Incoherently coupled dark-bright photorefractive solitons," Opt. Lett. 21, 1821-1823 (1996).
    [CrossRef] [PubMed]
  16. Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
    [CrossRef]
  17. Y. V. Kartashov and L. Torner, "Gray spatial solitons in nonlocal nonlinear media," Opt. Lett. 32, 946-948 (2007).
    [CrossRef] [PubMed]
  18. W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
    [CrossRef]
  19. W. Królikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
    [CrossRef]
  20. Z. H. Musslimani and J. Yang, "Transverse instability of strongly coupled dark bright Manakov vector solitons," Opt. Lett. 26, 1981-1983 (2001).
    [CrossRef]
  21. M. Lisak, A. Höök and D. Anderson, "Symbiotic solitary-wave pairs sustained by cross-phase modulation in optical fibers," J. Opt. Soc. Am. B 7, 810-814 (1990).
    [CrossRef]

2007 (1)

2006 (2)

S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
[CrossRef]

2005 (3)

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171-3173 (2005).
[CrossRef] [PubMed]

2004 (1)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

2003 (1)

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

2002 (2)

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

2001 (2)

Z. H. Musslimani and J. Yang, "Transverse instability of strongly coupled dark bright Manakov vector solitons," Opt. Lett. 26, 1981-1983 (2001).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

2000 (1)

W. Królikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

1997 (2)

A. W. Synder and D. J. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
[CrossRef]

A. P. Sheppard and Yu. S. Kivshar, "Polarized dark solitons in isotropic Kerr media," Phys. Rev. E 55, 4773-4782 (1997).
[CrossRef]

1996 (1)

1993 (1)

G. C. Duree, et al., "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

1992 (1)

M. Shalaby and A. J. Marthelemy, "Observation of the self-guided propagation of a dark and bright spatial solitonpair in a focusing nonlinear medium," IEEE J. Quantum Electron. 28, 2736-2741 (1992).
[CrossRef]

1990 (1)

1988 (1)

Afanasjev, V. V.

Anderson, D.

Assanto, G.

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

Bang, O.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Królikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

Brzdakiewicz, K. A.

Carmon, T.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Chen, Z.

Christodoulides, D. N.

Cohen, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Conti, C.

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

Coskun, T. H.

Desyatnikov, A. S.

Duree, G. C.

G. C. Duree, et al., "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

Edmundson, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Griesmaier, A.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Hensler, S.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Höök, A.

Joseph, R. I.

Kartashov, Y. V.

Kivshar, Yu. S.

Krolikowski, W.

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Królikowski, W.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Królikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

Lisak, M.

Lopez-Aguayo, S.

Manela, O.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Marthelemy, A. J.

M. Shalaby and A. J. Marthelemy, "Observation of the self-guided propagation of a dark and bright spatial solitonpair in a focusing nonlinear medium," IEEE J. Quantum Electron. 28, 2736-2741 (1992).
[CrossRef]

Mitchell, D. J.

A. W. Synder and D. J. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Musslimani, Z. H.

Neshev, D.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Nikolov, N. I.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

Peccianti, M.

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals," Opt. Lett. 27, 1460-1462 (2002).
[CrossRef]

Pfau, T.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Rasmussen, J. J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Rotschild, C.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Segev, M.

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

Z. Chen, M. Segev, T. H. Coskun, D. N. Christodoulides, Yu. S. Kivshar, and V. V. Afanasjev, "Incoherently coupled dark-bright photorefractive solitons," Opt. Lett. 21, 1821-1823 (1996).
[CrossRef] [PubMed]

Shalaby, M.

M. Shalaby and A. J. Marthelemy, "Observation of the self-guided propagation of a dark and bright spatial solitonpair in a focusing nonlinear medium," IEEE J. Quantum Electron. 28, 2736-2741 (1992).
[CrossRef]

Sheppard, A. P.

A. P. Sheppard and Yu. S. Kivshar, "Polarized dark solitons in isotropic Kerr media," Phys. Rev. E 55, 4773-4782 (1997).
[CrossRef]

Stuhler, J.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Synder, A. W.

A. W. Synder and D. J. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Torner, L.

Werner, J.

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Wyller, J.

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

Xu, Z.

Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
[CrossRef]

Z. Xu, Y. V. Kartashov, and L. Torner, "Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media," Opt. Lett. 30, 3171-3173 (2005).
[CrossRef] [PubMed]

Yang, J.

IEEE J. Quantum Electron. (1)

M. Shalaby and A. J. Marthelemy, "Observation of the self-guided propagation of a dark and bright spatial solitonpair in a focusing nonlinear medium," IEEE J. Quantum Electron. 28, 2736-2741 (1992).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons in nonlocal nonlinear media," J. Opt. B: Quantum Semiclassical Opt. 6, S288-S294 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. E (5)

A. P. Sheppard and Yu. S. Kivshar, "Polarized dark solitons in isotropic Kerr media," Phys. Rev. E 55, 4773-4782 (1997).
[CrossRef]

O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, "Collapse arrest and soliton stabilization in nonlocal nonlinear media," Phys. Rev. E 66, 046619 (2002).
[CrossRef]

Z. Xu, Y. V. Kartashov, and L. Torner, "Stabilization of vector soliton complexes in nonlocal nonlinear media," Phys. Rev. E 73, 055601 (2006).
[CrossRef]

W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, "Modulational instability in nonlocal nonlinear Kerr media," Phys. Rev. E 64, 016612 (2001).
[CrossRef]

W. Królikowski and O. Bang, "Solitons in nonlocal nonlinear media: Exact solutions," Phys. Rev. E 63, 016610 (2000).
[CrossRef]

Phys. Rev. Lett. (4)

G. C. Duree, et al., "Observation of self-trapping of an optical beam due to the photorefractive effect," Phys. Rev. Lett. 71, 533-536 (1993).
[CrossRef] [PubMed]

C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
[CrossRef] [PubMed]

C. Rotschild, O. Cohen, O. Manela, M. Segev, and T. Carmon, "Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons," Phys. Rev. Lett. 95, 213904 (2005).
[CrossRef] [PubMed]

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T. Pfau, "Bose-Einstein condensation of chromium," Phys. Rev. Lett. 94, 160401 (2005).
[CrossRef] [PubMed]

Science (1)

A. W. Synder and D. J. Mitchell, "Accessible Solitons," Science 276, 1538-1541 (1997).
[CrossRef]

Other (1)

Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, San Diego, 2003).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Intensity profiles of bright soliton (solid line), dark soliton (dashed line), and the refractive index (bold-dashed line) for the local (a) and nonlocal media (b) at the points A and B in (c), respectively. The bifurcation curves of fundamental dark-bright soliton pairs in local (d = 0, dashed line) and nonlocal media (d = 1, solid line) are shown in (c), where points A and B are marked with μu = 0.8 and μv = 1.

Fig. 2.
Fig. 2.

Relations between the propagation constant for bright soliton μu and the degree of nonlocality d at different fixed powers, P = 0.5 (solid line) and P = 1.0 (dashed line).

Fig. 3.
Fig. 3.

Threshold power of bright solitons versus the degree of nonlocality for μu = 0.8 and μv = 1.0. Solid and dashed lines are calculated by numerical and variational methods, respectively.

Equations (15)

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

i U ξ 1 2 2 U η 2 + n ( ξ , η ) U = 0 ,
i V ξ 1 2 2 V η 2 + n ( ξ , η ) V = 0 ,
n ( ξ , η ) = R ( η η ) ( U 2 + V 2 ) d η ,
R ( η ) = 1 2 d e η d ,
L = d η { i 2 ( U ξ U * U ξ * U ) + 1 2 U η 2 + 1 2 U 4 + U 2 V 2
+ d [ U η 2 + 1 2 U 2 ( U η η U * + U η η * U ) + U 2 2 η 2 V 2 ] } ,
n ( η ) = m = 0 1 m ! h m m U 2 η m
U 2 + V 2 + d [ 2 η 2 ( U 2 + V 2 ) ] ,
h m = i m d m d ω m H ( ω ) | ω = 0 ,
U ( η ) = A u sech ( η a u ) exp ( i ϕ u + i c u η 2 ) ,
V ( η ) = A v tanh ( η a v ) exp ( i ϕ v + i c v η 2 ) ,
P U 2 d η = 2 A u 2 a u ,
4 μ u 2 μ v 2 μ v 2 μ u + d [ 0.13 2 μ v 2 μ u + 0.877 2 μ v 2 μ u 1.57 ] .
U = e i μ u ξ [ u 0 + ( p u + i q u ) e λ ξ + ( p u * + i q u * ) e λ * ξ ] ,
V = e i μ v ξ [ v 0 + ( p v + i q v ) e λ ξ + ( p v * + i q v * ) e λ * ξ ] ,

Metrics