Abstract

We consider conditions of structural stability under which the array of singular beams preserves its topological structure and intensity distribution while slightly perturbing its intrinsic parameters. The orbital angular momentum of the array as a function of the array parameters is a characteristic function, and its extreme points correspond to stable and unstable array states.

© 2008 Optical Society of America

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  1. M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2001), Vol. 42, pp. 219-276.
    [CrossRef]
  2. V. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).
  3. V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
    [CrossRef]
  4. J. Lin, X.-C. Yuan, X. Peng, and H. B. Niu, "Deterministic approach to the generation of modified helical beams for optical manipulations," Opt. Express 13, 3862-3867 (2005).
    [CrossRef] [PubMed]
  5. A. Vaziri, G. Weihs, and A. Zelinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. A, Pure Appl. Opt. 4, S47-S51 (2002).
  6. L. Allen, S. M. Barnet, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
    [CrossRef]
  7. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solutions," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2005), Vol. 47, pp. 1-45.
    [CrossRef]
  8. C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
    [CrossRef]
  9. L. E. Helseth, "Optical vortices in focal region," Opt. Commun. 229, 85-91 (2004).
    [CrossRef]
  10. M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A, Pure Appl. Opt. 6, 259-268 (2004).
    [CrossRef]
  11. A. E. Siegman, Lasers (University Science, 1986).
  12. F. Gori, G. Gauttari, and C. Padovani, "Bessel-Gaussian beams," Opt. Commun. 64, 491-495 (1987).
    [CrossRef]
  13. A. P. Kiselev, "The localized light waves: paraxial and exact solution to the wave equation," Opt. Spectrosc. 102, 697-717 (2007).
    [CrossRef]
  14. J. C. Gutierrez-Vega and M. A. Bandress, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  15. M. Guizar-Sicairos and J. C. Gutierrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical system," Opt. Lett. 31, 2912-2914 (2006).
    [CrossRef] [PubMed]
  16. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, "Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express 13, 1749-1760 (2005).
    [CrossRef] [PubMed]
  17. E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
    [CrossRef]
  18. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
    [CrossRef] [PubMed]
  19. I. Freund, "Critical point explosions in two-dimensional wave fields," Opt. Commun. 159, 99-117 (1999).
    [CrossRef]
  20. I. Freund, "Saddle point wave fields," Opt. Commun. 163, 230-242 (1999).
    [CrossRef]
  21. J. Courtial, R. Zambrini, M. R. Dennis, and M. Vasnetsov, "Angular momentum of optical vortex array," Opt. Express 14, 938-949 (2006).
    [CrossRef] [PubMed]
  22. B. Lü and H. Ma, "Beam combination of radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
    [CrossRef]
  23. I. D. Maleev and G. A. Swartzlander, "Composite optical vortices," J. Opt. Soc. Am. B 20, 1169-1176 (2003).
    [CrossRef]
  24. Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
    [CrossRef]
  25. Ya. Cai and O. Lin, "Decentered elliptical Gaussian beams," Appl. Opt. 41, 4336-4340 (2002).
    [CrossRef] [PubMed]
  26. C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
    [CrossRef]
  27. Z. Bouchal and J. Courtial, "The connection of singular and nondiffractive optics," J. Opt. A, Pure Appl. Opt. 6, S184-S188 (2004).
    [CrossRef]
  28. S. Vyas and P. Senthsilkumaran, "Interferometric vortex array generator," Appl. Opt. 46, 2893-2898 (2007).
    [CrossRef] [PubMed]
  29. E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 174, 1273-1300 (2004), http://www.ufn.ru/ufn04/ufn04̱12/ufn0412a.pdf.
  30. E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams: optical and quantum aspects," Opt. Commun. 125, 302-323 (1996).
    [CrossRef]
  31. E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams," Opt. Commun. 102, 336-350 (1993).
    [CrossRef]
  32. T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Pitman, 1978).
  33. M. V. Berry, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XVIII, E.Wolf, ed. (Elsevier North-Holland, 1980), pp. 257-346.
    [CrossRef]
  34. I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. Lett. 50, 5164-5172 (1994).
  35. Ya. Izdebskaya, V. Shvedov, and A. Volyar, "Generation of higher-order optical vortices by a dielectric wedge," Opt. Lett. 30, 2472-2474 (2005).
    [CrossRef] [PubMed]
  36. X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
    [CrossRef]
  37. M. Abramowitz and I. StegunHandbook of Mathematical Functions (Dover, 1970).
  38. A. Volyar, V. Shvedov, Ya. Izdebskaya, T. Fadeyeva, and A. Rubass, "Structure and orbital angular momentum of singular array of Gaussian beams," Ukr. J. Phys. Opt. 3, 79-88 (2006), www.ifo.lviv.ua.
  39. K. O'Holleran, M. J. Padgett, and M. R. Dennis, "Topology of optical vortex lines formed by the interference of three, four and five plane waves," Opt. Express 14, 3039-3044 (2006).
    [CrossRef] [PubMed]
  40. Ya. Izdebskaya, T. Fadeyeva, V. Shvedov, and A. Volyar, "Vortex-bearing array of singular beams with very high orbital angular momentum," Opt. Lett. 31, 2523-2525 (2006).
    [CrossRef] [PubMed]
  41. M. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).
  42. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 2: Special Functions (Gordon and Breach, 1986).

2007 (4)

A. P. Kiselev, "The localized light waves: paraxial and exact solution to the wave equation," Opt. Spectrosc. 102, 697-717 (2007).
[CrossRef]

V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007).
[CrossRef] [PubMed]

S. Vyas and P. Senthsilkumaran, "Interferometric vortex array generator," Appl. Opt. 46, 2893-2898 (2007).
[CrossRef] [PubMed]

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

2006 (6)

2005 (5)

2004 (7)

Z. Bouchal and J. Courtial, "The connection of singular and nondiffractive optics," J. Opt. A, Pure Appl. Opt. 6, S184-S188 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 174, 1273-1300 (2004), http://www.ufn.ru/ufn04/ufn04̱12/ufn0412a.pdf.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

L. E. Helseth, "Optical vortices in focal region," Opt. Commun. 229, 85-91 (2004).
[CrossRef]

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A, Pure Appl. Opt. 6, 259-268 (2004).
[CrossRef]

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

2003 (2)

2002 (3)

Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
[CrossRef]

Ya. Cai and O. Lin, "Decentered elliptical Gaussian beams," Appl. Opt. 41, 4336-4340 (2002).
[CrossRef] [PubMed]

A. Vaziri, G. Weihs, and A. Zelinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. A, Pure Appl. Opt. 4, S47-S51 (2002).

2001 (1)

M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2001), Vol. 42, pp. 219-276.
[CrossRef]

2000 (1)

B. Lü and H. Ma, "Beam combination of radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

1999 (3)

V. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).

I. Freund, "Critical point explosions in two-dimensional wave fields," Opt. Commun. 159, 99-117 (1999).
[CrossRef]

I. Freund, "Saddle point wave fields," Opt. Commun. 163, 230-242 (1999).
[CrossRef]

1998 (1)

M. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).

1996 (1)

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams: optical and quantum aspects," Opt. Commun. 125, 302-323 (1996).
[CrossRef]

1994 (1)

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. Lett. 50, 5164-5172 (1994).

1993 (1)

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams," Opt. Commun. 102, 336-350 (1993).
[CrossRef]

1987 (1)

F. Gori, G. Gauttari, and C. Padovani, "Bessel-Gaussian beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

1986 (2)

A. E. Siegman, Lasers (University Science, 1986).

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 2: Special Functions (Gordon and Breach, 1986).

1980 (1)

M. V. Berry, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XVIII, E.Wolf, ed. (Elsevier North-Holland, 1980), pp. 257-346.
[CrossRef]

1978 (1)

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Pitman, 1978).

1970 (1)

M. Abramowitz and I. StegunHandbook of Mathematical Functions (Dover, 1970).

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 174, 1273-1300 (2004), http://www.ufn.ru/ufn04/ufn04̱12/ufn0412a.pdf.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams: optical and quantum aspects," Opt. Commun. 125, 302-323 (1996).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams," Opt. Commun. 102, 336-350 (1993).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. StegunHandbook of Mathematical Functions (Dover, 1970).

Ahluwalia, B. P. S.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

Allen, L.

L. Allen, S. M. Barnet, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Alonzo, C. A.

Bandress, M. A.

Barnet, S. M.

L. Allen, S. M. Barnet, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

Berry, M.

M. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).

Berry, M. V.

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A, Pure Appl. Opt. 6, 259-268 (2004).
[CrossRef]

M. V. Berry, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XVIII, E.Wolf, ed. (Elsevier North-Holland, 1980), pp. 257-346.
[CrossRef]

Bouchal, Z.

Z. Bouchal and J. Courtial, "The connection of singular and nondiffractive optics," J. Opt. A, Pure Appl. Opt. 6, S184-S188 (2004).
[CrossRef]

Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
[CrossRef]

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 2: Special Functions (Gordon and Breach, 1986).

Bu, J.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

Burge, R. E.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

Cai, Ya.

Cheong, W. C.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

Courtial, J.

J. Courtial, R. Zambrini, M. R. Dennis, and M. Vasnetsov, "Angular momentum of optical vortex array," Opt. Express 14, 938-949 (2006).
[CrossRef] [PubMed]

Z. Bouchal and J. Courtial, "The connection of singular and nondiffractive optics," J. Opt. A, Pure Appl. Opt. 6, S184-S188 (2004).
[CrossRef]

Cristobal, G.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Cuo, C.-S.

C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

Dändliker, R.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

Dennis, M. R.

Desyatnikov, A. S.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solutions," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2005), Vol. 47, pp. 1-45.
[CrossRef]

Dholakia, K.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Ding, J.-P.

C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

Fadeyeva, T.

A. Volyar, V. Shvedov, Ya. Izdebskaya, T. Fadeyeva, and A. Rubass, "Structure and orbital angular momentum of singular array of Gaussian beams," Ukr. J. Phys. Opt. 3, 79-88 (2006), www.ifo.lviv.ua.

Ya. Izdebskaya, T. Fadeyeva, V. Shvedov, and A. Volyar, "Vortex-bearing array of singular beams with very high orbital angular momentum," Opt. Lett. 31, 2523-2525 (2006).
[CrossRef] [PubMed]

Fernandez-Nieves, A.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Freund, I.

I. Freund, "Saddle point wave fields," Opt. Commun. 163, 230-242 (1999).
[CrossRef]

I. Freund, "Critical point explosions in two-dimensional wave fields," Opt. Commun. 159, 99-117 (1999).
[CrossRef]

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. Lett. 50, 5164-5172 (1994).

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Gauttari, G.

F. Gori, G. Gauttari, and C. Padovani, "Bessel-Gaussian beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Glückstad, J.

Gori, F.

F. Gori, G. Gauttari, and C. Padovani, "Bessel-Gaussian beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Guizar-Sicairos, M.

Gutierrez-Vega, J. C.

Han, Y.-J.

C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

Helseth, L. E.

L. E. Helseth, "Optical vortices in focal region," Opt. Commun. 229, 85-91 (2004).
[CrossRef]

Herzig, H. P.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

Ivanovskyy, A. A.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

Izdebskaya, Ya.

Khonina, S. N.

Kiselev, A. P.

A. P. Kiselev, "The localized light waves: paraxial and exact solution to the wave equation," Opt. Spectrosc. 102, 697-717 (2007).
[CrossRef]

Kivshar, Yu. S.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solutions," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2005), Vol. 47, pp. 1-45.
[CrossRef]

Kotlyar, V. V.

Lin, J.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

J. Lin, X.-C. Yuan, X. Peng, and H. B. Niu, "Deterministic approach to the generation of modified helical beams for optical manipulations," Opt. Express 13, 3862-3867 (2005).
[CrossRef] [PubMed]

Lin, O.

Lü, B.

B. Lü and H. Ma, "Beam combination of radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

Ma, H.

B. Lü and H. Ma, "Beam combination of radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

Maleev, I. D.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 2: Special Functions (Gordon and Breach, 1986).

McGloin, D.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Niu, H. B.

O'Holleran, K.

Padgett, M. J.

Padovani, C.

F. Gori, G. Gauttari, and C. Padovani, "Bessel-Gaussian beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Peng, X.

Poston, T.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Pitman, 1978).

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 2: Special Functions (Gordon and Breach, 1986).

Rockstuhl, C.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

Rodrigo, P. J.

Rubass, A.

A. Volyar, V. Shvedov, Ya. Izdebskaya, T. Fadeyeva, and A. Rubass, "Structure and orbital angular momentum of singular array of Gaussian beams," Ukr. J. Phys. Opt. 3, 79-88 (2006), www.ifo.lviv.ua.

Salt, M. G.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

Senthsilkumaran, P.

Shvartsman, N.

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. Lett. 50, 5164-5172 (1994).

Shvedov, V.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Skidanov, R. V.

Soifer, V. A.

Soskin, M. S.

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2001), Vol. 42, pp. 219-276.
[CrossRef]

Spalding, G. C.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Staliunas, K.

V. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).

Stegun, I.

M. Abramowitz and I. StegunHandbook of Mathematical Functions (Dover, 1970).

Stewart, I.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Pitman, 1978).

Summers, M. D.

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

Swartzlander, G. A.

Tao, S. H.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

Torner, L.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solutions," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2005), Vol. 47, pp. 1-45.
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2001), Vol. 42, pp. 219-276.
[CrossRef]

Vasnetsov, V.

V. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).

Vaziri, A.

A. Vaziri, G. Weihs, and A. Zelinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. A, Pure Appl. Opt. 4, S47-S51 (2002).

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 174, 1273-1300 (2004), http://www.ufn.ru/ufn04/ufn04̱12/ufn0412a.pdf.

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams: optical and quantum aspects," Opt. Commun. 125, 302-323 (1996).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams," Opt. Commun. 102, 336-350 (1993).
[CrossRef]

Volyar, A.

Vyas, S.

Wang, H.-T.

C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zelinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. A, Pure Appl. Opt. 4, S47-S51 (2002).

Yuan, X.-C.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

J. Lin, X.-C. Yuan, X. Peng, and H. B. Niu, "Deterministic approach to the generation of modified helical beams for optical manipulations," Opt. Express 13, 3862-3867 (2005).
[CrossRef] [PubMed]

Zambrini, R.

Zelinger, A.

A. Vaziri, G. Weihs, and A. Zelinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. A, Pure Appl. Opt. 4, S47-S51 (2002).

Zhang, L.-S.

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

Zhang, Y.

C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B: Lasers Opt. (1)

X.-C. Yuan, B. P. S. Ahluwalia, S. H. Tao, W. C. Cheong, L.-S. Zhang, J. Lin, J. Bu, and R. E. Burge, "Wavelength-scalable micro-fabricated wedge for generation of optical vortex beam in optical manipulation," Appl. Phys. B: Lasers Opt. 86, 209-213 (2007).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (5)

M. V. Berry, "Optical vortices evolving from helicoidal integer and fractional phase steps," J. Opt. A, Pure Appl. Opt. 6, 259-268 (2004).
[CrossRef]

Z. Bouchal and J. Courtial, "The connection of singular and nondiffractive optics," J. Opt. A, Pure Appl. Opt. 6, S184-S188 (2004).
[CrossRef]

A. Vaziri, G. Weihs, and A. Zelinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. A, Pure Appl. Opt. 4, S47-S51 (2002).

V. Garces-Chavez, D. McGloin, M. D. Summers, A. Fernandez-Nieves, G. C. Spalding, G. Cristobal, and K. Dholakia, "The reconstruction of optical angular momentum after distortion in amplitude, phase and polarization," J. Opt. A, Pure Appl. Opt. 6, S235-S238 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Generalized Gaussian beams," J. Opt. A, Pure Appl. Opt. 6, S157-S161 (2004).
[CrossRef]

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

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

Opt. Commun. (9)

C.-S. Cuo, Y. Zhang, Y.-J. Han, J.-P. Ding, and H.-T. Wang, "Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering," Opt. Commun. 259, 449-454 (2006).
[CrossRef]

I. Freund, "Critical point explosions in two-dimensional wave fields," Opt. Commun. 159, 99-117 (1999).
[CrossRef]

I. Freund, "Saddle point wave fields," Opt. Commun. 163, 230-242 (1999).
[CrossRef]

B. Lü and H. Ma, "Beam combination of radial laser array: Hermite-Gaussian model," Opt. Commun. 178, 395-403 (2000).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams: optical and quantum aspects," Opt. Commun. 125, 302-323 (1996).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral-type beams," Opt. Commun. 102, 336-350 (1993).
[CrossRef]

C. Rockstuhl, A. A. Ivanovskyy, M. S. Soskin, M. G. Salt, H. P. Herzig, and R. Dändliker, "High-resolution experiment of phase singularities produced by computer-generated holograms," Opt. Commun. 242, 163-169 (2004).
[CrossRef]

L. E. Helseth, "Optical vortices in focal region," Opt. Commun. 229, 85-91 (2004).
[CrossRef]

F. Gori, G. Gauttari, and C. Padovani, "Bessel-Gaussian beams," Opt. Commun. 64, 491-495 (1987).
[CrossRef]

Opt. Express (4)

Opt. Lett. (5)

Opt. Spectrosc. (1)

A. P. Kiselev, "The localized light waves: paraxial and exact solution to the wave equation," Opt. Spectrosc. 102, 697-717 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

I. Freund and N. Shvartsman, "Wave-field phase singularities: the sign principle," Phys. Rev. Lett. 50, 5164-5172 (1994).

Other (12)

M. Berry, "Paraxial beams of spinning light," Proc. SPIE 3487, 6-11 (1998).

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Volume 2: Special Functions (Gordon and Breach, 1986).

A. E. Siegman, Lasers (University Science, 1986).

M. Abramowitz and I. StegunHandbook of Mathematical Functions (Dover, 1970).

A. Volyar, V. Shvedov, Ya. Izdebskaya, T. Fadeyeva, and A. Rubass, "Structure and orbital angular momentum of singular array of Gaussian beams," Ukr. J. Phys. Opt. 3, 79-88 (2006), www.ifo.lviv.ua.

M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2001), Vol. 42, pp. 219-276.
[CrossRef]

V. Vasnetsov and K. Staliunas, eds., Optical Vortices, Vol. 228 of Horizons of World Physics (Nova Science, 1999).

L. Allen, S. M. Barnet, and M. J. Padgett, Optical Angular Momentum (IOP, 2003).
[CrossRef]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, "Optical vortices and vortex solutions," in Progress in Optics, E.Wolf, ed. (Elsevier North-Holland, 2005), Vol. 47, pp. 1-45.
[CrossRef]

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Pitman, 1978).

M. V. Berry, "Catastrophe optics: morphologies of caustics and their diffraction patterns," in Progress in Optics XVIII, E.Wolf, ed. (Elsevier North-Holland, 1980), pp. 257-346.
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, "Spiral light beams," Phys. Usp. 174, 1273-1300 (2004), http://www.ufn.ru/ufn04/ufn04̱12/ufn0412a.pdf.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the vortex-beam array.

Fig. 2
Fig. 2

Intensity distributions (I) and interference patterns (II) for the beam array in the state { N , l } at z = 0 plane.

Fig. 3
Fig. 3

Transformations of the intensity distribution of the beam array (I) and the spiral beam (II) in the state { 5 , 2 } along the z axis.

Fig. 4
Fig. 4

Vortex trajectories of the beam array in states (a) {3, 1} and (b) {4, 3} with R = 1 .

Fig. 5
Fig. 5

Double explosion of the critical point in a spiral beam in the state { 9 , 2 , 1 } with the intensity (I) and phase (II) distributions.

Fig. 6
Fig. 6

Evolution of the specific OAM and intensity distributions (a) for the simplest array of two Gaussian beams N = 2 for l = 0 (curve 1) and l = ± 1 (curve 2), R = 0.3 ; (b) for the beam array with { 2 l l } , R = 2 .

Fig. 7
Fig. 7

Evolution of a vortex-beam array (a) and a spiral beam (b) in the state {8, 1, 6}.

Fig. 8
Fig. 8

Specific OAM η ( α ¯ ) for a spiral beam in the state { 9 , 2 , 1 } .

Fig. 9
Fig. 9

Spiral beams with a zero OAM in the state { 8 , 3 , 3 } .

Equations (54)

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x n = x r 0 , y n = y cos α z sin α ,
z n = y sin α + z cos α ,
r n 2 r 2 2 r r 0 cos ϑ n 2 r α z sin ϑ n + r o 2 + α 2 z 2 ,
z n z ( 1 α 2 2 ) + α r sin ϑ n ,
( 2 x 2 + 2 y 2 2 i k z 2 ) Ψ ̃ n = 0 ,
Ψ ̃ n 1 Z n [ r exp ( i σ ϑ n ) C Z n ] M exp ( i k r n 2 2 Z n ) ,
Ψ n = Ψ ̃ n ( r , φ , z ) exp ( i k z n ) .
Ψ = n = 1 N Ψ n exp { i exp [ i φ n ( l + σ M ) ] } .
[ r exp ( i σ ϑ n ) C Z n ] M
= 1 Z M s = 0 M ( 1 ) M s ( M s ) C M s r s exp ( i σ s ϑ n ) .
exp ( i k r n 2 2 Z ) exp ( i k z n ) = Ψ 0 exp { A exp ( i ϑ n ) + B exp ( i ϑ n ) } exp [ i ( l + σ M ) φ n ] ,
Ψ 0 = exp ( k z 0 2 α 2 ) exp ( i k r 2 2 Z i k z ) exp [ i z 0 Z ( α ¯ 2 R 2 ) 2 ] ,
A = i z 0 Z ( R α ¯ ) r w 0 , B = i z 0 Z ( R + α ¯ ) r w 0 ,
R = r 0 w 0 , α ¯ = α α diff , α diff = 2 k w 0 .
exp [ x 2 [ t + 1 t ] ] = p = t p I p ( x ) ,
Ψ = Ψ 0 s = 0 M a s ( r Z ) s p = ( A B ) p 2 I p ( 2 A B ) exp [ i ( p + σ s ) φ ] n = 1 N exp [ i ( l + σ ( M s ) p ) φ n ] .
n = 1 N exp [ i ( l + σ M p σ s ) 2 π N n ] = { N if l + σ ( M s ) p = m N , m = 0 , ± 1 , ± 2 , , 0 elsewhere }
Ψ = N Ψ 0 m = exp { i ( m N + l + σ M ) φ } s = 0 M a s R s [ m N + l + σ ( M s ) ] 2 I m N + l + σ ( M s ) ( 2 ξ R ) ,
a s = ( M s ) ( C Z ) M s ( w 0 Z ) s = ( M s ) [ r 0 + i σ α z z + i z 0 ] M s ( w 0 Z ) s ,
= R α ¯ R + α ¯ , ξ = z 0 Z α ¯ 2 R 2 , R = r w 0 .
Ψ = N Ψ 0 m = exp { i ( m N + l ) φ } ( R α ¯ R + α ¯ ) ( m N + l ) 2 I m N + l ( 2 z 0 Z α ¯ 2 R 2 R ) .
Ψ N Ψ 0 m = exp { i ( m N + l ) φ } m N + l ! ( m N + l ) 2 ( ξ R ) m N + l .
Ψ N Ψ 0 ( z = 0 ) m = exp { i ( m N + l ) φ } m N + l ! ( m N + l ) 2 ( ξ ¯ R ) m N + l ,
Ψ 2 Ψ 0 [ m = 1 ( R + α ¯ ) 2 m 1 ( 2 m 1 ) ! R 2 m 1 exp { i ( 2 m 1 ) φ } + m = 0 ( R α ¯ ) 2 m + 1 ( 2 m + 1 ) ! R 2 m + 1 exp { i ( 2 m + 1 ) φ } ] = 2 Ψ 0 { sinh [ ( R + α ¯ ) R exp ( i φ ) ] + sinh [ ( R α ¯ ) R exp ( i φ ) ] } .
Ψ N Ψ 0 [ m = 1 ( R + α ¯ ) m N l ( m N l ) ! R m N l exp { i ( m N l ) φ } + m = 0 ( R α ¯ ) m N + l ( m N + l ) ! R m N + l exp { i ( m N + l ) φ } ] .
R l exp ( i l φ ) [ ( R α ¯ ) l l ! + ( R + α ¯ ) N l ( N l ) ! R N 2 l exp ( i N φ ) ] = 0 , N 2 l ,
R N l exp [ i ( N l ) φ ] [ ( R + α ¯ ) N l ( N l ) ! + ( R α ¯ ) l l ! R 2 l N exp ( i N φ ) ] = 0 , N < 2 l .
Q = { sign ( α ¯ ) ( N l ) , ( α ¯ l ) > 0 , structurally unstable spiral beams sign ( α ¯ ) l , ( α ¯ l ) < 0 , structurally stable spiral beams } , N 2 l , α ¯ = R ,
Q = { sign ( α ¯ ) ( N l ) , ( α ¯ l ) < 0 , structurally stable spiral beams sign ( α ¯ ) l , ( α ¯ l ) > 0 , structurally unstable spiral beams } , N < 2 l , α ¯ = R .
Ψ = N Ψ 0 { m = 1 exp { i ( m N l M ) φ } s = 0 M a s ( R + α ¯ ) m N ( l + M ) + s R m N ( l + M ) + 2 s ( m N l M + s ) ! + m = 0 R m N + l + M exp { i ( m N + l + M ) φ } s = 0 M a s ( R α ¯ ) m N + l + M s ( m N + l + M s ) ! } ,
a s = ( i w 0 z 0 ) M ( M s ) ( 1 ) s R M s .
Ψ N Ψ 0 ( i w 0 z 0 ) M { ( 1 ) N M R N M exp { i ( N M ) φ } ( 2 R ) N M R M ( N M ) ! + R l + M exp { i M φ } } .
R crit = M ! M 2 ,
Ψ 2 N Ψ 0 ( i w 0 z 0 ) M cos M φ .
φ j = 2 j + 1 M π 2 , j = 0 , 1 , , M ,
Ψ N Ψ 0 m = 1 exp { i ( m N l M ) φ } s = 0 M a s ( R α ¯ ) m N l ( M s ) R m N l M + 2 s ( m N l M + s ) ! + { m = 0 R m N + l + M exp { i ( m N + l + M ) φ } ϴ m N , l , M ( R , α ¯ + R ) } ,
ϴ m N , l , M ( R , α ¯ + R ) = s = 0 M a s ( R ) ( R + α ) m N + l + M s ( m N + l + M s ) ! .
Ψ R l + M exp { i ( l + M ) φ } ϴ 0 , l , M ( R ) + R N + l + M exp { i ( N + l + M ) φ } ϴ N , l , M ( R ) .
Ψ = N Ψ 0 { m = 1 R m N ( l M ) exp { i ( m N ( l M ) ) φ } ϴ m N , l , M ( R , R + α ¯ ) + m = 0 exp { i ( m N + l M ) φ } s = 0 M a s ( R α ¯ ) m N + l ( M s ) [ m N + l ( M s ) ] ! R m N + l M + 2 s } ,
ϴ m N , l , M ( R , α ¯ ) = ( i w 0 z 0 ) M s = 0 M a s ( α ¯ + R ) m N l + M s ( m N l + M s ) ! .
ϴ 0 , l , M ( M l ) = s = 0 M l a s ( R + α ¯ ) M ( l + s ) ( M l s ) ! .
Q = sign ( α ¯ ) [ N ( l + σ M ) ] , ( α ¯ l ) > 0 ,
( σ l ) > 0 , N 2 l + σ M .
L z = Im Ψ φ Ψ .
0 x β 1 exp ( p x 2 ) I μ ( c x ) I ν ( c x ) d x = 1 2 ( c 2 ) μ + ν 1 ( p ) β + μ + ν Γ [ ( β + μ + ν ) 2 ] Γ ( μ + 1 ) Γ ( ν + 1 ) × F 3 3 ( μ + ν + 1 2 , μ + ν 2 + 1 , μ + ν + β 2 ; μ + 1 , ν + 1 , μ + ν + 1 ; c 2 p ) , β + μ + ν > 0 ,
L z = w 0 2 2 Ψ ̃ 0 2 m = 1 ( m N l σ M ) s , s = 0 M a s a s ( R + α ¯ ) p m , s Γ [ ( p m , s + s + s + 2 ) 2 ] ν ! μ ! F 3 3 ( { p m , s } ; 4 ( R 2 α ¯ 2 ) ) + w 0 2 2 Ψ ̃ 0 2 m = 0 ( m N + l + σ M ) s , s = 0 M a s a s 2 q m , s ( R α ¯ ) q m , s Γ [ ( q m , s + s + s + 2 ) 2 ] ν + ! μ + ! F 3 3 ( { q m , s } ; 4 ( R 2 α ¯ 2 ) ) , for l + σ ( M s ) 0 , N l + σ M .
{ p m , s } = p m , s + 1 2 , p m , s 2 + 1 , p m , s + s + s + 2 2 ; μ + 1 , ν + 1 , μ + ν + 1 ,
{ q m , s } = q m , s + 1 2 , q m , s 2 + 1 , q m , s + s + s + 2 2 ; μ + + 1 , ν + + 1 , μ + + ν + 1 .
L z = w 0 2 2 Ψ ̃ 0 2 m = 1 ( m N l σ M ) s , s = 0 M a s a s Γ ( p m , s + s + s + 2 2 ) 2 p m , s ν ! μ ! ( 2 R ) p m , s .
L z = 1 2 Ψ ̃ 0 exp ( R 2 α ¯ 2 ) m = ( m N + l ) ( R α ¯ R + α ¯ ) m N + l I m N + l ( 2 [ R 2 α ¯ 2 ] ) .
η = L z Ψ Ψ .
η σ M R α ¯ .
η = ( α ¯ R ) coth ( R 2 + α ¯ 2 2 ) , l = ± 1 ,
η = ( α ¯ R ) tanh ( R 2 + α ¯ 2 2 ) , l = 0 .

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