Abstract

We define a new parameter for Laguerre–Gaussian (LG) beams, named Qp,l, which is only related to mode indices p and l. This parameter is able to distinguish and evaluate LG beams. The Qp,l values are first calculated theoretically and then measured experimentally for several different LG beams. Another mode quality parameter, M2 values, are also measured. The comparison between Qp,l and M2 shows the same trend for the quality of the LG mode, while the measurement of Qp,l is much easier than M2.

© 2013 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]
  2. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
    [CrossRef]
  3. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004).
    [CrossRef]
  4. M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
    [CrossRef]
  5. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
    [CrossRef]
  6. M. S. S. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
    [CrossRef]
  7. G. Brand, Am. J. Phys. 67, 55 (1999).
    [CrossRef]
  8. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
    [CrossRef]
  9. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
    [CrossRef]
  10. M. Harris, C. A. Hill, P. R. Tapster, and J. M. Vaughan, Phys. Rev. A 49, 3119 (1994).
    [CrossRef]
  11. M. J. Padgett, J. Arlt, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996).
    [CrossRef]
  12. H. I. Sztul and R. R. Alfano, Opt. Lett. 31, 999 (2006).
    [CrossRef]
  13. G. C. G. Berkhout and M. W. Beijersbergen, Phys. Rev. Lett. 101, 100801 (2008).
    [CrossRef]
  14. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, Phys. Rev. Lett. 105, 053904 (2010).
    [CrossRef]
  15. T. Kaiser, D. Flamm, S. Schöter, and M. Duparré, Opt. Express 17, 9347 (2009).
    [CrossRef]
  16. L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
    [CrossRef]
  17. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, Phys. Rev. Lett. 88, 257901 (2002).
    [CrossRef]
  18. G. C. Berkhout, M. P. Lavery, J. Courtial, M. W. Beijersbergen, and M. J. Padgett, Phys. Rev. Lett. 105, 153601 (2010).
    [CrossRef]
  19. M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
    [CrossRef]
  20. S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, Opt. Lett. 36, 4398 (2011).
    [CrossRef]
  21. : Test methods for laser beam widths, divergence angles and beam propagation ratios—Part 1,2 (2005).
  22. J. Arlt, “Applications of the Laguerre-Gaussian beams and Bessel beams to both non linear optics and atom optics,” Ph.D thesis (School of Physics and Astronomy, University of St Andrews, 1999).

2012 (1)

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

2011 (2)

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
[CrossRef]

S. Prabhakar, A. Kumar, J. Banerji, and R. P. Singh, Opt. Lett. 36, 4398 (2011).
[CrossRef]

2010 (2)

G. C. Berkhout, M. P. 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]

2009 (1)

2008 (2)

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

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

2006 (1)

2004 (1)

2002 (1)

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

2001 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

1999 (1)

G. Brand, Am. J. Phys. 67, 55 (1999).
[CrossRef]

1996 (2)

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef]

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

1995 (1)

M. S. S. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

1994 (2)

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

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

1992 (1)

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

Alfano, R. R.

Allen, L.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

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

Arlt, J.

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

J. Arlt, “Applications of the Laguerre-Gaussian beams and Bessel beams to both non linear optics and atom optics,” Ph.D thesis (School of Physics and Astronomy, University of St Andrews, 1999).

Banerji, J.

Barnett, S. M.

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

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

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

Basistiy, M. S. S. I. V.

M. S. S. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Beijersbergen, M. W.

G. C. Berkhout, M. P. 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]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

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

Berkhout, G. C.

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

Berkhout, G. C. G.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
[CrossRef]

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

Brand, G.

G. Brand, Am. J. Phys. 67, 55 (1999).
[CrossRef]

Cao, M.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[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]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Courtial, J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
[CrossRef]

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

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

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

Duparré, M.

Enger, J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef]

Flamm, D.

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]

Franke-Arnold, S.

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

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

Friese, M. E. J.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef]

Gao, H.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Gao, S.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Gibson, G.

Guo, W.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Han, L.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Harris, M.

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

Heckenberg, N. R.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef]

Hickmann, J. M.

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

Hill, C. A.

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

Kaiser, T.

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

Kumar, A.

Kwiat, P. G.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

Lavery, M. P.

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

Lavery, M. P. J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
[CrossRef]

Leach, J.

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

Li, F.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Liu, H.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Liu, R.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Padgett, M. J.

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
[CrossRef]

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

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

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

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

Pas’ko, V.

Prabhakar, S.

Rubinsztein-Dunlop, H.

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef]

Schöter, S.

Simpson, N.

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

Singh, R. P.

Soares, W. C.

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

Soskin, M. S.

M. S. S. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[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]

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]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

Vanderveen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

M. S. S. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Vaughan, J. M.

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

Wei, D.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

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

Zhang, P.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

Am. J. Phys. (2)

G. Brand, Am. J. Phys. 67, 55 (1999).
[CrossRef]

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

Europhys. Lett. (1)

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, Europhys. Lett. 99, 34003 (2012).
[CrossRef]

J. Opt. (1)

M. P. J. Lavery, G. C. G. Berkhout, J. Courtial, and M. J. Padgett, J. Opt. 13, 064006 (2011).
[CrossRef]

Nat. Phys. (1)

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, Nat. Phys. 4, 282 (2008).
[CrossRef]

Opt. Commun. (3)

M. W. Beijersbergen, L. Allen, H. E. L. O. Vanderveen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, Opt. Commun. 112, 321 (1994).
[CrossRef]

M. S. S. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, Opt. Commun. 119, 604 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (3)

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

M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, Phys. Rev. A 54, 1593 (1996).
[CrossRef]

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

Phys. Rev. Lett. (5)

G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601 (2001).
[CrossRef]

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

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

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

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

Other (2)

: Test methods for laser beam widths, divergence angles and beam propagation ratios—Part 1,2 (2005).

J. Arlt, “Applications of the Laguerre-Gaussian beams and Bessel beams to both non linear optics and atom optics,” Ph.D thesis (School of Physics and Astronomy, University of St Andrews, 1999).

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Figures (4)

Fig. 1.
Fig. 1.

Relative intensity distribution of LG beams with p = 0 , l = 1 .

Fig. 2.
Fig. 2.

Theoretical Q p , l values of LG beams with different p and l .

Fig. 3.
Fig. 3.

Measured Q p , l values in the x direction (a) and y direction (b) of beams with p = 0 and l = 1 , 2, 3 [pictures are shown in the middle of Figs. (a) and (b)]. The straight lines are the theoretical values.

Fig. 4.
Fig. 4.

Normalized intensity distribution in the x direction. The pattern of LG beam ( p = 0 and l = 1 ) was taken at far-field.

Tables (1)

Tables Icon

Table 1. Numerically Calculated Q p , l Values

Equations (6)

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

I = u p l * u p l = 2 p ! π ( p + | l | ) ! 1 ω 2 ( 2 ( x / ω ) 2 ) | l | [ L p | l | ( 2 ( x / ω ) 2 ) ] 2 e 2 ( x / ω ) 2 = 2 p ! π ( p + | l | ) ! 1 ω 2 f p l ( t ) ,
f p l ( t ) = ( 2 t 2 ) | l | [ L p | l | ( 2 t 2 ) ] 2 e 2 t 2 .
Q p , l = x 1 x 2 x 1 .
d I d x = 2 p ! π ( p + | l | ) ! 1 ω 3 d f p l ( t ) d ( t ) 1 ω = 0 .
1 2 = I half I m = 2 p ! π ( p + | l | ) ! 1 ω 2 f p l ( t ) 2 p ! π ( p + | l | ) ! 1 ω 2 f p l ( t m ) = f p l ( t ) f p l ( t m ) .
Q p , l = x 1 x 2 x 1 .

Metrics