Abstract

We comment on a recent paper by D. Ling et al. [Appl. Opt. 45, 4102 (2006)]. In that paper, the authors adopted the entire matrix formalism that we established in a previous work [J. Opt. Soc. Am. A 22, 1909 (2005)] for finding the eigenmodes of an unstable Bessel resonator. Nevertheless, the results are inaccurate mainly because (a) it was overlooked that light crosses through the axicon twice in a complete round trip and (b) the numerical method used to evaluate the diffraction integral equations cannot resolve the eigenvalues and eigenfields for the given resonator configuration.

© 2007 Optical Society of America

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References

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  1. D. Ling, C. Li, and J. Li, "Eigenfields and output beams of an unstable Bessel-Gauss resonator," Appl. Opt. 45, 4102-4108 (2006).
    [CrossRef] [PubMed]
  2. R. I. Hernández-Aranda, S. Chávez-Cerda, and J. C. Gutiérrez-Vega, "Theory of the unstable Bessel resonator," J. Opt. Soc. Am. A 22, 1909-1917 (2005).
    [CrossRef]
  3. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, "Bessel-Gauss resonator with spherical output mirror: geometrical and wave-optics analysis," J. Opt. Soc. Am. A 20, 2113-2122 (2003).
    [CrossRef]
  4. M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, "Construction and characterization of CO2 laser with an axicon based Bessel-Gauss resonator," in Laser Resonators and Beam Control VIII, Al. V. Kudryashov and Al. H. Paxton, eds., Proc SPIE 5708, 323-331 (2005).
  5. J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
    [CrossRef]
  6. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axicon-based Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
    [CrossRef]
  7. P. Muys and E. Vandamme, "Direct generation of Bessel beams," Appl. Opt. 41, 6375-6379 (2002).
    [CrossRef] [PubMed]
  8. C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
    [CrossRef]
  9. A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell. Syst. Tech. J. 40, 453-488 (1961).
  10. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004).
    [CrossRef]
  11. A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, 1992).
  13. R. L. Sanderson and W. Streifer, "Comparison of laser mode calculations," Appl. Opt. 8, 131-136 (1969).
    [CrossRef] [PubMed]
  14. W. D. Murphy and M. L. Bernabe, "Numerical procedures for solving nonsymmetric eigenvalue problems associated with optical resonators," Appl. Opt. 17, 2358-2365 (1978).
    [CrossRef] [PubMed]
  15. W. P. Latham Jr. and G. C. Dente, "Matrix methods for bare resonator eigenvalue analysis," Appl. Opt. 19, 1618-1621 (1980).
    [CrossRef] [PubMed]

2006

2005

R. I. Hernández-Aranda, S. Chávez-Cerda, and J. C. Gutiérrez-Vega, "Theory of the unstable Bessel resonator," J. Opt. Soc. Am. A 22, 1909-1917 (2005).
[CrossRef]

M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, "Construction and characterization of CO2 laser with an axicon based Bessel-Gauss resonator," in Laser Resonators and Beam Control VIII, Al. V. Kudryashov and Al. H. Paxton, eds., Proc SPIE 5708, 323-331 (2005).

2004

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004).
[CrossRef]

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

2003

2002

2001

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, "Axicon-based Bessel resonator: analytical description and experiment," J. Opt. Soc. Am. A 18, 1986-1992 (2001).
[CrossRef]

1980

1978

1969

1961

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell. Syst. Tech. J. 40, 453-488 (1961).

Alvarez, M.

M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, "Construction and characterization of CO2 laser with an axicon based Bessel-Gauss resonator," in Laser Resonators and Beam Control VIII, Al. V. Kudryashov and Al. H. Paxton, eds., Proc SPIE 5708, 323-331 (2005).

Bernabe, M. L.

Buchter, S. C.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Chávez-Cerda, S.

Dente, G. C.

Elfstrom, H.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, 1992).

Fox, A. G.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell. Syst. Tech. J. 40, 453-488 (1961).

Guizar-Sicairos, M.

M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, "Construction and characterization of CO2 laser with an axicon based Bessel-Gauss resonator," in Laser Resonators and Beam Control VIII, Al. V. Kudryashov and Al. H. Paxton, eds., Proc SPIE 5708, 323-331 (2005).

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields," J. Opt. Soc. Am. A 21, 53-58 (2004).
[CrossRef]

Gutiérrez-Vega, J. C.

Hakola, A.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Hernández-Aranda, R. I.

Kajava, T.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Katranji, E. G.

Khilo, A. N.

Latham, W. P.

Li, C.

Li, J.

Li, T.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell. Syst. Tech. J. 40, 453-488 (1961).

Ling, D.

Murphy, W. D.

Muys, P.

New, G. H. C.

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

Paakkonen, P.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, 1992).

Rodríguez-Masegosa, R.

M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, "Construction and characterization of CO2 laser with an axicon based Bessel-Gauss resonator," in Laser Resonators and Beam Control VIII, Al. V. Kudryashov and Al. H. Paxton, eds., Proc SPIE 5708, 323-331 (2005).

J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, "Bessel-Gauss resonator with spherical output mirror: geometrical and wave-optics analysis," J. Opt. Soc. Am. A 20, 2113-2122 (2003).
[CrossRef]

Rogel-Salazar, J.

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

Ryzhevich, A. A.

Sanderson, R. L.

Simonen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Streifer, W.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, 1992).

Tsangaris, C. L.

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

Turunen, J.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

Vandamme, E.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, 1992).

Appl. Opt.

Bell. Syst. Tech. J.

A. G. Fox and T. Li, "Resonant modes in a maser interferometer," Bell. Syst. Tech. J. 40, 453-488 (1961).

J. Opt. Soc. Am. A

Opt. Commun.

A. Hakola, S. C. Buchter, T. Kajava, H. Elfstrom, J. Simonen, P. Paakkonen, and J. Turunen, "Bessel-Gauss output beam from a diode-pumped Nd:YAG laser," Opt. Commun. 238, 335-340 (2004).

J. Rogel-Salazar, G. H. C. New, and S. Chávez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190, 117-122 (2001).
[CrossRef]

C. L. Tsangaris, G. H. C. New, and J. Rogel-Salazar, "Unstable Bessel beam resonator," Opt. Commun. 223, 233-238 (2003).
[CrossRef]

Proc SPIE

M. Alvarez, M. Guizar-Sicairos, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, "Construction and characterization of CO2 laser with an axicon based Bessel-Gauss resonator," in Laser Resonators and Beam Control VIII, Al. V. Kudryashov and Al. H. Paxton, eds., Proc SPIE 5708, 323-331 (2005).

Other

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: the Art of Scientific Computing (Cambridge U. Press, 1992).

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Equations (25)

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

T ( r ) = exp ( i k θ 0 r ) ,
θ 0 = ( n 1 ) α
T 2 ( r ) = exp ( i 2 k θ 0 r )
M dax = [ 1 0 2 θ 0 / r 1 ] ,
M dax
A = 1 2 θ 0 L / r 1
T 1
a 1
a 2
E 1 ( r 1 )
E 2 ( r 2 ) = ( i ) n + 1 k e i k L B 1 0 a 1 E 1 ( r 1 ) J n ( k r 1 r 2 B 1 ) × exp [ i k 2 B 1 ( A 1 r 1 2 + D 1 r 2 2 ) ] r 1 d r 1 .
T 1
exp { i k 2 L [ ( 1 2 θ 0 L r 1 ) r 1 2 + r 2 2 ] } = exp ( i k θ 0 r 1 ) × exp [ i k 2 L ( r 1 2 + r 2 2 ) ] .
E 2
E 2 ( r 2 ) = ( i ) n + 1 k e i k L L 0 a 1 [ E 1 ( r 1 ) exp ( i k θ 0 r 1 ) ] J n ( k r 1 r 2 L ) × exp [ i k 2 L ( r 1 2 + r 2 2 ) ] r 1 d r 1 .
E 1 ( r 1 ) exp ( i k θ 0 r 1 )
exp ( i k θ 0 r 1 )
L = a 2 ( n 1 ) α ,
k t
J n ( k t r )
k t = k  sin [ ( n 1 ) α ]
k t = 7241 m 1
0.43   mm
[ 0 < r < 5   mm ]
M = 256

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