Abstract

We introduce a new class of elliptically modulated self-trapped singular beams in isotropic nonlinear media where nonlocality plays a crucial role in their existence. The analytical expressions in the highly nonlocal nonlinear limit of these elliptically shaped self-trapped beams, or ellipticons, is obtained and their existence in more general nonlocal nonlinear media is demonstrated. We show that the ellipticons represent a generalization of several known self-trapped beams, for example vortex solitons, azimuthons, and the Hermite and Laguerre solitons clusters. For the limit of the highly nonlocal nonlinear medium, the ellipticons are described in close form in terms of the InceGauss functions.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003) p. 540.
  2. A. W. Snyder and D. J. Mitchell, "Accessible solitons," Science 276, 1538-1541 (1997).
    [CrossRef]
  3. E. A. Ultanir, G. Stegeman, C. H. Lange, and F. Lederer, "Coherent interactions of dissipative spatial solitons," Opt. Lett. 29, 283-285 (2004).
    [CrossRef] [PubMed]
  4. C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
    [CrossRef]
  5. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  6. 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]
  7. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
    [CrossRef] [PubMed]
  8. A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev E 71, 065603 (2005).
    [CrossRef]
  9. S. Lopez-Aguayo, A. S. Desyatnikov, Yu. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
    [CrossRef] [PubMed]
  10. 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]
  11. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
    [CrossRef] [PubMed]
  12. C. Conti, M. Peccianti and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
    [CrossRef] [PubMed]
  13. S. Lopez-Aguayo, A. S. Desyatnikov, and Yu. S. Kivshar, "Azimuthons in nonlocal nonlinear media," Opt. Express 14, 7903-7908 (2006).
    [CrossRef] [PubMed]
  14. A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
    [CrossRef] [PubMed]
  15. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
    [CrossRef] [PubMed]

2007 (1)

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (3)

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev E 71, 065603 (2005).
[CrossRef]

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. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

2004 (2)

E. A. Ultanir, G. Stegeman, C. H. Lange, and F. Lederer, "Coherent interactions of dissipative spatial solitons," Opt. Lett. 29, 283-285 (2004).
[CrossRef] [PubMed]

C. Conti, M. Peccianti and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

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 (1)

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]

1999 (1)

C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
[CrossRef]

1997 (1)

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

1992 (1)

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

Assanto, G.

C. Conti, M. Peccianti and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[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]

Bang, O.

S. Lopez-Aguayo, A. S. Desyatnikov, Yu. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

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]

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

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]

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, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[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]

Crosignani, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

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

S. Lopez-Aguayo, A. S. Desyatnikov, Yu. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Dreischuh, A.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Fischer, B.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

Gerton, J. M.

C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
[CrossRef]

Hulet, R. C.

C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
[CrossRef]

Kivshar, Yu.

Kivshar, Yu. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

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

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev E 71, 065603 (2005).
[CrossRef]

Krolikowski, W.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

S. Lopez-Aguayo, A. S. Desyatnikov, Yu. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
[CrossRef] [PubMed]

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]

Lange, C. H.

Lederer, F.

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]

Mitchell, D. J.

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

Neshev, D. N.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Peccianti, M.

C. Conti, M. Peccianti and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[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]

Petersen, D. E.

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

Rasmussen, J. J.

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]

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]

Sackett, C. A.

C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
[CrossRef]

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]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

Skupin, S.

Snyder, A. W.

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

Stegeman, G.

Sukhorukov, A. A.

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

Ultanir, E. A.

Weilling, M.

C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
[CrossRef]

Wyller, J.

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]

Yakimenko, A. I.

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev E 71, 065603 (2005).
[CrossRef]

Yariv, A.

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

Zaliznyak, Y. A.

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev E 71, 065603 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev E (1)

A. I. Yakimenko, Y. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev E 71, 065603 (2005).
[CrossRef]

Phys. Rev. E (1)

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]

Phys. Rev. Lett. (8)

A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Krolikowski, "Observation of attraction between dark solitons," Phys. Rev. Lett. 96, 043901 (2006).
[CrossRef] [PubMed]

C. A. Sackett, J. M. Gerton, M. Weilling, and R. C. Hulet, "Measurements of collective collapse in a Bose-Einstein condensate with attractive interactions," Phys. Rev. Lett. 82, 876-879 (1999).
[CrossRef]

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]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992).
[CrossRef] [PubMed]

C. Conti, M. Peccianti and G. Assanto, "Observation of optical spatial solitons in a highly nonlocal medium," Phys. Rev. Lett. 92, 113902 (2004).
[CrossRef] [PubMed]

A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, "Azimuthons: Spatially modulated vortex solitons," Phys. Rev. Lett. 95, 203904 (2005).
[CrossRef] [PubMed]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Yu. S. Kivshar, "Laguerre and Hermite Soliton clusters in nonlocal nonlinear media," Phys. Rev. Lett. 98, 053901 (2007).
[CrossRef] [PubMed]

Science (1)

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

Other (1)

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003) p. 540.

Supplementary Material (11)

» Media 1: AVI (1365 KB)     
» Media 2: AVI (1406 KB)     
» Media 3: AVI (1339 KB)     
» Media 4: AVI (1520 KB)     
» Media 5: AVI (1537 KB)     
» Media 6: AVI (2087 KB)     
» Media 7: AVI (2097 KB)     
» Media 8: AVI (537 KB)     
» Media 9: AVI (1459 KB)     
» Media 10: AVI (1374 KB)     
» Media 11: AVI (454 KB)     

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 (7)

Fig. 1.
Fig. 1.

(1.3 Mb) (a) Intensity and phase of a stationary accessible ellipticon given by Ψ+ 5,3(ξ,η) for ε=0,1, and ∞. (b) Phase structure around the interfocal line. (c) Intensity and phase distribution of a stationary accessible ellipticon with parameters p=5 and m=5 in HNN media. The intensity pattern remain invariant in propagation while the phase front rotate around the interfocal line. [Media 1]

Fig. 2.
Fig. 2.

(a) Nodal lines for the pure even and the pure odd part of Ψ+5,3(ξ,η) for ε=1 and ∞ respectively. Red and blue lines represent nodal lines of the even and odd components, respectively. Positive and negative vortices are represented by white and black circles, respectively. (b) Orbital angular momentum carried by Ψ+ p,m in function of ε for p=2n+m and m={1,2, …,7}.

Fig. 3.
Fig. 3.

Different scenarios for the propagation of accessible rotating ellipticons given by the combination of two stationary accessible ellipticons: (a) (1.4 Mb) self-imaging phenomenon in the case of p 1=6, m 1=2, p 2=2, and m 2=2, (b) (1.3 Mb) stationary behavior; p 1=3, m 1=1, p 2=3, and m 2=3, (c) (1.5 Mb) rotation of the intensity pattern and the phase front in opposite directions, here p 1=8, m 1=4, p 2=6, and m 2=6, and (d) (1.5 Mb) rotation of the intensity pattern and the phase front in the same direction; p 1=8, m 1=4, p 2=10, and m 2=10. In all the cases ε=1 and L=2π/a. [Media 2] [Media 3][Media 4] [Media 5]

Fig. 4.
Fig. 4.

Propagation dynamics of an elliptcon with parameters p=4, m=4, ε=1, and P 0=103 in an xy box of 2.6×2.6 in a nonlocal nonlinear medium. (a) Using directly the accessible ellipticon solution the beam diffracts and the maximum normalized intensity decays [see (b)]. (c) Using the same trial function but modified with the variational approach (A=1.6066) the beam remain self-trapped and the maximum normalized intensity oscillates remaining within a finite and small range [see (d)].

Fig. 5.
Fig. 5.

Intensity and phase of two ellipticons in nonlocal nonlinear media with parameters (a) (2 Mb) p=2, m=2, ε=0.75, and P0=103 in an xy box of 4.2×4.2, and (b) (2.1 Mb) p=3, m=3, ε=1, and P0=103 in an xy box of 4.8×4.8.

Fig. 6.
Fig. 6.

(a) Propagation in nonlocal nonlinear media of a vortex soliton of single charge and double ring (or soliton Laguerre mode L 11). (b) Different modes that also coexist in the propagation before mentioned.

Fig. 7.
Fig. 7.

(a) (1.4 Mb) Elliptically intensity-rotating beam with ε=0 close to the HNN limit (P0=106) in a xy box of 0.8×0.8 produced by two ellipticons with parameters p1=10, m1=10, p2=2, and m2=2. (b) (1.4 Mb) Intensity rotating beam produced by three ellipticons with parameters p1=5, m1=5, p2=5, m 2=3, p 3=5, and m 3=1 close to the HNN limit (P 0=106) in a xy box of 0.8×0.8, here ε=1, for all the three beams. [Media 8] [Media 11]

Equations (15)

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

2 i k n 0 z E ( r , z ) + n 0 2 E ( r , z ) + 2 k 2 n ( I , z ) E ( r , z ) = 0 ,
n ( I , z ) = R ( r r ) E ( r , z ) 2 d r ,
R ( r ) = 1 π σ 2 exp ( r 2 σ 2 ) ,
R ( r r ' ) = R 0 + ( r r ) · R 0 + 1 2 [ ( r r ) · ] 2 R 0 + ,
n ( I , z ) P 0 π σ 2 ( 1 r 2 σ 2 ) ,
2 i k z U ( r , z ) + 2 U ( r , z ) k 2 a 2 r 2 U ( r , z ) = 0 ,
Ψ p , m ± ( ξ , η ) = [ C C p m ( i ξ ; ε ) C p m ( η ; ε ) ± i S S p m ( i ξ ; ε ) S p m ( η ; ε ) ] exp ( a k r 2 2 ) ,
β = ( p + 1 ) a ,
P 0 = π n 0 k 2 σ 4 ( p + 1 ) 2 β 2 .
J z = h ¯ r × Im ( U * U ) d x d y U 2 d x d y ,
Φ ( ξ , η ) = A 1 Ψ p 1 , m 1 + ( ξ , η ) + A 2 Ψ p 2 , m 2 + ( ξ , η ) ,
p p = τ ( m m ) ,
ω = a τ .
= 2 k n 0 Γ E ( r ) 2 n 0 E ( r ) 2 + k 2 E ( r ) 2 2 R ( r r ) E ( r ) 2 d r ,
A = P 0 2 ( p + 1 ) π n 0 P 0 3 2 k 2 + ( π σ 2 n 0 2 k 2 ) Ψ p , m ± ( ξ , η ) 2 d r Ψ p , m ± ( ξ , η ) 2 [ exp ( r r ' 2 σ 2 ) Ψ p , m ± ( ξ ' , η ' ) 2 d r ' ] d r .

Metrics