Abstract

We demonstrate that spatial nonlocal response provides an effective physical mechanism for stabilization of recently introduced azimuthally modulated self-trapped rotating singular optical beams or azimuthons [see A. S. Desyatnikov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 95, 203904 (2005)]. We find that stable azimuthons become possible when the nonlocality parameter exceeds a certain threshold value and, in a sharp contrast to local media, the azimuthons with N peaks can also exist for N < 2m, where m is the azimuthon topological charge.

© 2006 Optical Society of America

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  1. Yu. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003), p. 540.
  2. D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
    [CrossRef] [PubMed]
  3. C. Conti, M. Peccianti, and G. Assanto, "Route to nonlocality and observation of accessible solitons," Phys. Rev. Lett. 91, 073901 (2003).
    [CrossRef] [PubMed]
  4. 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]
  5. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons, and beam propagation in spatially nonlocal nonlinear media," J. Opt. B. Quantum Semiclassical Opt. 6, S288-S294 (2004).
    [CrossRef]
  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. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, "Nonlocal spatial soliton interactions in nematic liquid crystals", Opt. Lett. 27, 1460-1462 (2002).
    [CrossRef]
  8. N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
    [CrossRef] [PubMed]
  9. 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]
  10. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," in Prog. Opt. 47, Ed. E. Wolf (North-Holland, Amsterdam, 2005), pp. 291-391.
    [CrossRef]
  11. A. I. Yakimenko, Yu. A. Zaliznyak, and Yu. S. Kivshar, "Stable vortex solitons in nonlocal self-focusing nonlinear media," Phys. Rev. E 71, 065603 (2005).
    [CrossRef]
  12. D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
    [CrossRef] [PubMed]
  13. A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton slusters," Phys. Rev. Lett. 88, 053901 (2002).
    [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. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, "Stable rotating dipole solitons in nonlocal optical media," Opt. Lett. 31, 1100-1102 (2006).
    [CrossRef] [PubMed]
  16. B. A. Malomed, "Variational methods in nonlinear fiber optics and related fields," Prog. Opt. 43, E. Wolf, ed., (North-Holland, Amsterdam, 2002), p.71-191.
    [CrossRef]

2006 (1)

2005 (5)

D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (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]

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

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)

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

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (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 (3)

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]

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton slusters," Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

1993 (1)

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

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.

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

D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons, and beam propagation in spatially 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]

Blasberg, T.

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Briedis, D.

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]

Christiansen, P. L.

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]

Desyatnikov, A. S.

S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. 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]

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton slusters," Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Edmundson, D.

D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

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

Kartashov, Y. V.

Kivshar, Y. S.

Kivshar, Yu. S.

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

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton slusters," Phys. Rev. Lett. 88, 053901 (2002).
[CrossRef] [PubMed]

Krolikowski, W.

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

D. Briedis, D. E. Petersen, D. Edmundson, W. Krolikowski, and O. Bang, "Ring vortex solitons in nonlocal nonlinear media," Opt. Express 13, 435-443 (2005).
[CrossRef] [PubMed]

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons, and beam propagation in spatially 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]

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]

Neshev, D.

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
[CrossRef] [PubMed]

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

Nikolov, N. I.

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

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
[CrossRef] [PubMed]

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]

Petersen, D. E.

Rasmussen, J. J.

N. I. Nikolov, D. Neshev, W. Krolikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, "Attraction of nonlocal dark optical solitons," Opt. Lett. 29, 286-288 (2004).
[CrossRef] [PubMed]

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons, and beam propagation in spatially 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]

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]

Skupin, S.

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]

Suter, D.

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Torner, L.

Wyller, J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, "Modulational instability, solitons, and beam propagation in spatially 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]

Xu, Z.

Yakimenko, A. I.

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

Zaliznyak, Yu. A.

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

J. Opt. B. (1)

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

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

D. Suter and T. Blasberg, "Stabilization of transverse solitary waves by a nonlocal response of the nonlinear medium," Phys. Rev. A 48, 4583-4587 (1993).
[CrossRef] [PubMed]

Phys. Rev. E (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]

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

Phys. Rev. Lett. (4)

A. S. Desyatnikov and Yu. S. Kivshar, "Rotating optical soliton slusters," Phys. Rev. Lett. 88, 053901 (2002).
[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]

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]

Other (3)

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," in Prog. Opt. 47, Ed. E. Wolf (North-Holland, Amsterdam, 2005), pp. 291-391.
[CrossRef]

B. A. Malomed, "Variational methods in nonlinear fiber optics and related fields," Prog. Opt. 43, E. Wolf, ed., (North-Holland, Amsterdam, 2002), p.71-191.
[CrossRef]

Yu. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003), p. 540.

Supplementary Material (10)

» Media 1: AVI (2029 KB)     
» Media 2: AVI (1394 KB)     
» Media 3: AVI (1766 KB)     
» Media 4: AVI (2020 KB)     
» Media 5: AVI (2031 KB)     
» Media 6: AVI (2223 KB)     
» Media 7: AVI (2222 KB)     
» Media 8: AVI (2254 KB)     
» Media 9: AVI (2223 KB)     
» Media 10: AVI (2196 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

(a) Profiles R(r) of single-charge azimuthons (m = 1) with n = 0 (vortex soliton), and with n = 0.5 for N = 2,3. Contours n = const for azimuthons with m = 1 for N = 2 (b) and N = 3 (c). Existence domains for azimuthons with topological charges m = 1 (d), 2 (e), and 3 (f), for different values of N. The domain for N = 2 is shaded as an example.

Fig. 2.
Fig. 2.

Propagation of the azimuthons with m = 1, N = 3, n = 0.5 and different k. (a,b) Breakup for (a) k = 2.3 and ω = 1.2 and (b) k = 0.7 and ω = -0.25. (c) (2 Mb) Unstable azimuthon with ω = 40 that remains trapped by a strong nonlocal potential at k = 143.4. (d) (1.4 Mb) Stable azimuthon with ω = -9 and k = 91.7. Top row - variational solutions at z = 0, and bottom row - after propagation of (a,b) z = 8.9, (c) z = 7.4, and (d) z = 120.

Fig. 3.
Fig. 3.

Dynamics of (a) (1.8 Mb) unstable azimuthon with m = 1, N = 4, k = 0.5, n = 0.5, and ω = -0.5, and (b) (2 Mb) stable azimuthon with n = 0.5,k = 115.3, and ω = -17.3. (c) (2 Mb) Propagation of a double-charge azimuthon with n = 0.5, k = 200, and ω = 0.0025. (d) (2.2 Mb) Propagation of a triple-charge azimuthon with n = 0.5, k = 173.9 and ω = 9. Top row: variational solutions; bottom row: after propagation of (a) z = 8.1, (b) z = 110, and (c,d) z = 10.

Fig. 4.
Fig. 4.

(a) (2.2 Mb) Propagation of the azimuthon with m = 3, N = 2, n = 0.5, k = 257.5, and ω = 19.5. Examples of higher-order azimuthons with m = 6 and n = 0.5, for the cases: (b) (2.3 Mb) N = 5, k = 645.7, ω = 24.9; (c) (2.2 Mb) N = 6, k = 624.3, ω = 21.3; and (d) (2.2 Mb) N = 7, k = 603, and ω = 17.8. Top: variational solutions; middle: after propagation of (a) z = 10 and (b-d) z = 2; bottom: corresponding phases.

Equations (7)

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

i E z + Δ E + E K ( r ρ ) E ( ρ ) 2 d ρ = 0 ,
K ( r ) = ( 1 π σ 2 ) exp ( r 2 σ 2 ) ,
V ( r , θ ) = R ( r ) [ 1 n sin 2 ( N θ 2 ) ] 1 / 2 exp { i ψ ( θ ) } ,
ψ ( θ ) = ω ω 0 θ + 2 N ( m ω ω 0 ) tan 1 [ 1 n tan ( N θ 2 ) ] ,
d 2 R d r 2 + 1 r d R d r g 2 r 2 R k 0 R + R N ( R 2 , r ) = 0 ,
N ( R 2 , r ) = π ( 2 n ) e r 2 0 ρ d ρ e ρ 2 R 2 ( ρ ) { I 0 ( 2 r ρ ) + n 2 2 ( 2 n ) 2 I N ( 2 r ρ ) } ,
Ω ± = m ± N 2 4 1 n F N , F N = 0 ρ d ρ e ρ 2 R 2 ( ρ ) 0 r d r e r 2 R 2 ( r ) I N ( 2 ρ r ) .

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