Abstract

We analyzed the spatial spectrum of the diffraction intensity pattern of an ideal Bessel beam and found an implicit rule that the number of the bright rings in the spatial spectrum is equal to the topological charge of the Bessel beam. The radius of the bright and dark rings has some relation with the topological charge and can be determined accurately. It provides us with a new way for measuring the topological charge of an optical vortex through its diffraction intensity pattern after an axicon. The results of simulation coincide with the theory.

© 2011 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  2. K. T. Gahagan and G. A. Swartzlander Jr., Opt. Lett. 21, 827 (1996).
    [CrossRef] [PubMed]
  3. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
    [CrossRef]
  4. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, Opt. Lett. 24, 608 (1999).
    [CrossRef]
  5. A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
    [CrossRef] [PubMed]
  6. Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
    [CrossRef]
  7. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef] [PubMed]
  8. J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
    [CrossRef]
  9. K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).
    [CrossRef] [PubMed]
  10. M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
    [CrossRef] [PubMed]
  11. M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2011 (2)

2010 (2)

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

2009 (3)

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

C.-S. Guo, L.-L. Lu, and H.-T. Wang, Opt. Lett. 34, 3686 (2009).
[CrossRef] [PubMed]

2008 (1)

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

2006 (1)

2004 (4)

K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

2003 (1)

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

2002 (3)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

1999 (1)

1996 (2)

K. T. Gahagan and G. A. Swartzlander Jr., Opt. Lett. 21, 827 (1996).
[CrossRef] [PubMed]

M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
[CrossRef]

1994 (1)

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Alfano, R. R.

Allen, L.

M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Anderson, M. E.

Arit, J.

M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
[CrossRef]

Barnett, S.

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Beijersbergen, M. W.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Berkhout, G. C. G.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

Bouchal, Z.

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

Castillo, S.

Celechovsky, R.

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

Chávez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Courtial, J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

de Araujo, L. E. E.

Dholakia, K.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Fonseca, E. J. S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Gahagan, K. T.

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Gibson, G.

Grier, D. G.

K. Ladavac and D. G. Grier, Opt. Express 12, 1144 (2004).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Guo, C.-S.

C.-S. Guo, L.-L. Lu, and H.-T. Wang, Opt. Lett. 34, 3686 (2009).
[CrossRef] [PubMed]

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Haist, T.

Harris, M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Hickmann, J. M.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Jennewein, T.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Khonina, N.

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Kotlyar, V. V.

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Ladavac, K.

Lavery, M. P. J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Lin, Q.

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Lu, L.-L.

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Padgett, M.

Padgett, M. J.

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Pan, J. W.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Pas’ko, V.

Rativa, D.

Reicherter, M.

Shinkarev, M. V.

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

Simpson, N.

M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
[CrossRef]

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Soifer, V. A.

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

Swartzlander, G. A.

Sztul, H. I.

Tapster, P. R.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Tiziani, H. J.

Uspleniev, G. V.

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

Vasnetsov, M.

Vaughan, J. M.

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Vaziri, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Vohnsen, B.

Wagemann, E. U.

Wang, H.-T.

Wang, Z.

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Wei, G.-X.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Weihs, G.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Yue, S.-J.

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

Zeilinger, A.

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

Zhang, Z.

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Am. J. Phys. (1)

M. Padgett, J. Arit, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
[CrossRef]

Appl. Phys. Lett. (1)

C.-S. Guo, S.-J. Yue, and G.-X. Wei, Appl. Phys. Lett. 94, 231104 (2009).
[CrossRef]

J. Mod. Opt. (1)

N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, J. Mod. Opt. 39, 1147 (1992).
[CrossRef]

J. Opt. A (1)

Z. Wang, Z. Zhang, and Q. Lin, J. Opt. A 11, 085702 (2009).
[CrossRef]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, Nature 419, 145 (2002).
[CrossRef] [PubMed]

New J. Phys. (1)

Z. Bouchal and R. Celechovsky, New J. Phys. 6, 131 (2004).
[CrossRef]

Opt. Commun. (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, Opt. Commun. 207, 169 (2002).
[CrossRef]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (6)

A. Vaziri, J. W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601(2004).
[CrossRef] [PubMed]

G. C. G. Berkhout, M. P. J. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
[CrossRef]

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
[CrossRef] [PubMed]

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1965).

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Figures (4)

Fig. 1
Fig. 1

Spatial spectrum of intensity distribution of ideal Bessel beams at (a) second order and (b) fiftth order.

Fig. 2
Fig. 2

Schematic of the experiment setup.

Fig. 3
Fig. 3

Results of simulation with l = 2 . (a1)–(a3) Normalized diffraction intensity pattern at z = 200 mm , 1000 mm , and 1600 mm . (b1)–(b3) Spatial spectrum of (a1)–(a3).

Fig. 4
Fig. 4

Results of simulation with l = 5 . (a1)–(a3) Normalized diffraction intensity pattern at z = 400 mm , 1000 mm , and 2000 mm . (b1)–(b3) Spatial spectrum of (a1)–(a3).

Equations (7)

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

u l ( r , φ ) = J l ( β r ) exp ( i l φ ) ,
I l ( r ) = | J l ( β r ) | 2 = J l 2 ( β r ) .
g ( ρ , θ ) = FT [ J l 2 ( β r ) ] = 1 λ f 0 J l 2 ( β r ) r d r 0 2 π exp [ j 2 π r ρ λ f cos ( φ θ ) ] d φ = 1 λ f 0 J l 2 ( β r ) J 0 ( 2 π r ρ λ f ) r d r ,
g ( ρ , θ ) = 1 2 π β 2 λ f cos [ ( 2 l + 1 ) arcsin ( π ρ β λ f ) ] + π ρ β λ f sin [ ( 2 l + 1 ) arcsin ( π ρ β λ f ) ] π ρ β λ f [ 1 ( π ρ β λ f ) 2 ] 1 / 2 = 1 2 π β 2 λ f cos { ( 2 l + 1 ) arcsin ( π ρ β λ f ) arcsin [ π ρ β λ f ( 1 + ( π ρ β λ f ) 2 ) 1 / 2 ] } π ρ β λ f [ 1 ( π ρ β λ f ) 4 ] 1 / 2 .
( 2 l + 1 ) arcsin ( π ρ β λ f ) arcsin [ π ρ β λ f ( 1 + ( π ρ β λ f ) 2 ) 1 / 2 ] = ( n + 1 2 ) π ,
( 2 l + 1 ) arcsin ( π ρ β λ f ) arcsin [ π ρ β λ f ( 1 + ( π ρ β λ f ) 2 ) 1 / 2 ] = n π .
u ( r , φ ) = C ( 2 r ω ) l exp ( r 2 ω 2 ) exp ( i l φ ) ,

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