Abstract

A variant of the Gaussian beam expansion method consists in expanding the Bessel function J0 appearing in the Fresnel–Kirchhoff integral into a finite sum of complex Gaussian functions to derive an analytical expression for a Laguerre–Gaussian beam diffracted through a hard-edge aperture. However, the validity range of the approximation depends on the number of expansion coefficients that are obtained by optimization–computation directly. We propose another solution consisting in expanding J0 onto a set of collimated Laguerre–Gaussian functions whose waist depends on their number and then, depending on its argument, predicting the suitable number of expansion functions to calculate the integral recursively.

© 2009 Optical Society of America

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References

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  1. R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier Science, 2001), pp. 325-386.
    [CrossRef]
  2. A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962 (2005).
    [CrossRef] [PubMed]
  3. N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).
  4. G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
    [CrossRef]
  5. R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).
  6. H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J.Fox, ed. (Polytechnic Press, Polytechnic Institute of Brooklyn, 1964), pp. 333-347.
  7. K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
    [CrossRef]
  8. N. S. Petrovic and A. D. Rakic, “Modeling diffraction in free space optical interconnects by the mode expansion method,” Appl. Opt. 42, 5308-5318 (2003).
    [CrossRef] [PubMed]
  9. G. Toker, A. Brunfeld, and J. Shamir, “Diffraction of apertured Gaussian beams: solution by expansion in Chebyshev polynomials,” Appl. Opt. 32, 4706-4712 (1993).
    [CrossRef] [PubMed]
  10. J. J. Snyder, “Modeling laser beam diffraction and propagation by the mode-expansion method,” Appl. Opt. 46, 5056-5061 (2007).
    [CrossRef] [PubMed]
  11. G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423-429 (1996).
    [CrossRef]
  12. N. S. Petrovic, “Modelling diffraction in optical interconnects,” Ph.D. thesis (University of Queensland, School of Information Technology and Electrical Engineering, 2004).
  13. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  14. H.-J. Kim, L. W. Schmerr, and A. Sedov, “Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview,” J. Acoust. Soc. Am. 119, 1971-1978 (2006).
    [CrossRef] [PubMed]
  15. W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008) (abstract).
    [CrossRef]
  16. D. Ding, X. Tong, and P. He, “Supplementary notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 118, 608-611 (2005).
    [CrossRef]
  17. J. H. Huang and D. Desheng, “An alternative analytical description of truncated Gaussian beams,” J. Opt. A, Pure Appl. Opt. 10, 015003 (2008).
    [CrossRef]
  18. R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
    [CrossRef]
  19. R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
    [CrossRef]
  20. M. Brunel, D. Mgharaz, and S. Coetmellec, “Generation of femtosecond diffraction-compensated beam through an opaque disk,” Opt. Express 16, 10390-10397 (2008).
    [CrossRef] [PubMed]
  21. Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
    [CrossRef]
  22. E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
    [CrossRef]
  23. Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic Proceedings, N.Burgess and L.Ciminiera, eds. (IEEE, 2001), pp. 155-162.

2008

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008) (abstract).
[CrossRef]

J. H. Huang and D. Desheng, “An alternative analytical description of truncated Gaussian beams,” J. Opt. A, Pure Appl. Opt. 10, 015003 (2008).
[CrossRef]

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

M. Brunel, D. Mgharaz, and S. Coetmellec, “Generation of femtosecond diffraction-compensated beam through an opaque disk,” Opt. Express 16, 10390-10397 (2008).
[CrossRef] [PubMed]

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

2007

2006

H.-J. Kim, L. W. Schmerr, and A. Sedov, “Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview,” J. Acoust. Soc. Am. 119, 1971-1978 (2006).
[CrossRef] [PubMed]

2005

2004

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

2003

1996

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423-429 (1996).
[CrossRef]

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

1993

1988

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1972

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Ait-Ameur, K.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

Aït-Ameur, K.

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Bailey, D. H.

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic Proceedings, N.Burgess and L.Ciminiera, eds. (IEEE, 2001), pp. 155-162.

Borghi, R.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Brunel, M.

Brunfeld, A.

Cagniot, E.

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Coetmellec, S.

Davidson, N.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962 (2005).
[CrossRef] [PubMed]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier Science, 2001), pp. 325-386.
[CrossRef]

de Doncker-Kapenga, E.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

de Saint Denis, R.

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

Derrar-Kaddour, Z.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Desheng, D.

J. H. Huang and D. Desheng, “An alternative analytical description of truncated Gaussian beams,” J. Opt. A, Pure Appl. Opt. 10, 015003 (2008).
[CrossRef]

Ding, D.

D. Ding, X. Tong, and P. He, “Supplementary notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 118, 608-611 (2005).
[CrossRef]

Friesem, A. A.

A. A. Ishaaya, N. Davidson, and A. A. Friesem, “Very high-order pure Laguerre-Gaussian mode selection in a passive Q-switched Nd:YAG laser,” Opt. Express 13, 4952-4962 (2005).
[CrossRef] [PubMed]

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier Science, 2001), pp. 325-386.
[CrossRef]

Fromager, M.

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Fukumitsu, O.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Hasman, E.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier Science, 2001), pp. 325-386.
[CrossRef]

He, P.

D. Ding, X. Tong, and P. He, “Supplementary notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 118, 608-611 (2005).
[CrossRef]

Hida, Y.

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic Proceedings, N.Burgess and L.Ciminiera, eds. (IEEE, 2001), pp. 155-162.

Huang, J. H.

J. H. Huang and D. Desheng, “An alternative analytical description of truncated Gaussian beams,” J. Opt. A, Pure Appl. Opt. 10, 015003 (2008).
[CrossRef]

Ishaaya, A. A.

Jackel, S.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Ji, P.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008) (abstract).
[CrossRef]

Kahaner, D. K.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Kim, H.-J.

H.-J. Kim, L. W. Schmerr, and A. Sedov, “Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview,” J. Acoust. Soc. Am. 119, 1971-1978 (2006).
[CrossRef] [PubMed]

Kogelnik, H.

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J.Fox, ed. (Polytechnic Press, Polytechnic Institute of Brooklyn, 1964), pp. 333-347.

Lenz, G.

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423-429 (1996).
[CrossRef]

Li, X. S.

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic Proceedings, N.Burgess and L.Ciminiera, eds. (IEEE, 2001), pp. 155-162.

Liu, W.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008) (abstract).
[CrossRef]

Lumer, Y.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Machavariani, G.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Martel, G.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Meir, A.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Mgharaz, D.

Moshe, I.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Oron, R.

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier Science, 2001), pp. 325-386.
[CrossRef]

Passilly, N.

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

Petrovic, N. S.

N. S. Petrovic and A. D. Rakic, “Modeling diffraction in free space optical interconnects by the mode expansion method,” Appl. Opt. 42, 5308-5318 (2003).
[CrossRef] [PubMed]

N. S. Petrovic, “Modelling diffraction in optical interconnects,” Ph.D. thesis (University of Queensland, School of Information Technology and Electrical Engineering, 2004).

Piessens, R.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Rakic, A. D.

Santarsiero, M.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Schmerr, L. W.

H.-J. Kim, L. W. Schmerr, and A. Sedov, “Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview,” J. Acoust. Soc. Am. 119, 1971-1978 (2006).
[CrossRef] [PubMed]

Sedov, A.

H.-J. Kim, L. W. Schmerr, and A. Sedov, “Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview,” J. Acoust. Soc. Am. 119, 1971-1978 (2006).
[CrossRef] [PubMed]

Shamir, J.

Shibukawa, M.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Snyder, J. J.

Taleb, A.

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

Tanaka, K.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

Toker, G.

Tong, X.

D. Ding, X. Tong, and P. He, “Supplementary notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 118, 608-611 (2005).
[CrossRef]

Überhuber, C. W.

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Yang, J.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008) (abstract).
[CrossRef]

Appl. Opt.

IEEE Trans. Microwave Theory Tech.

K. Tanaka, M. Shibukawa, and O. Fukumitsu, “Diffraction of a wave beam by an aperture,” IEEE Trans. Microwave Theory Tech. 20, 749-755 (1972).
[CrossRef]

J. Acoust. Soc. Am.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

H.-J. Kim, L. W. Schmerr, and A. Sedov, “Generation of the basis sets for multi-Gaussian ultrasonic beam models--an overview,” J. Acoust. Soc. Am. 119, 1971-1978 (2006).
[CrossRef] [PubMed]

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008) (abstract).
[CrossRef]

D. Ding, X. Tong, and P. He, “Supplementary notes on the Gaussian beam expansion,” J. Acoust. Soc. Am. 118, 608-611 (2005).
[CrossRef]

J. Mod. Opt.

N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagation factor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51, 2279-2286 (2004).

J. Opt. A, Pure Appl. Opt.

J. H. Huang and D. Desheng, “An alternative analytical description of truncated Gaussian beams,” J. Opt. A, Pure Appl. Opt. 10, 015003 (2008).
[CrossRef]

Opt. Commun.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncation series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Diffraction properties of opaque disks outside and inside a laser cavity,” Opt. Commun. 281, 444-451 (2008).
[CrossRef]

G. Lenz, “Far-field diffraction of truncated higher-order Laguerre-Gaussian beams,” Opt. Commun. 123, 423-429 (1996).
[CrossRef]

Z. Derrar-Kaddour, A. Taleb, K. Ait-Ameur, G. Martel, and E. Cagniot, “Alternative model for computing intensity patterns through apertured ABCD systems,” Opt. Commun. 281, 1384-1395 (2008).
[CrossRef]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Aït-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281, 4449-4454 (2008).
[CrossRef]

Opt. Express

Opt. Mater.

G. Machavariani, N. Davidson, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “New methods of mode conversion and beam brightness enhancement in a high-power laser,” Opt. Mater. 30, 1723-1730 (2008).
[CrossRef]

Other

R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner, Quadpack. A Subroutine Package for Automatic Integration (Springer Verlag, 1983).

H. Kogelnik, “Coupling and conversion coefficients for optical modes,” in Proceedings of the Symposium on Quasi-Optics, J.Fox, ed. (Polytechnic Press, Polytechnic Institute of Brooklyn, 1964), pp. 333-347.

N. S. Petrovic, “Modelling diffraction in optical interconnects,” Ph.D. thesis (University of Queensland, School of Information Technology and Electrical Engineering, 2004).

R. Oron, N. Davidson, E. Hasman, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” in Progress in Optics, Vol. 42, E.Wolf, ed. (Elsevier Science, 2001), pp. 325-386.
[CrossRef]

Y. Hida, X. S. Li, and D. H. Bailey, “Algorithms for quad-double precision floating point arithmetic,” in 15th IEEE Symposium on Computer Arithmetic Proceedings, N.Burgess and L.Ciminiera, eds. (IEEE, 2001), pp. 155-162.

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Figures (4)

Fig. 1
Fig. 1

Off-axis intensity distributions of a Gaussian beam diffracted by a hard-edge aperture and measured at N f = 8 .

Fig. 2
Fig. 2

Off-axis intensity distributions of a collimated TEM 20 beam diffracted by a hard-edge aperture with a thin lens and measured at the focal plane.

Fig. 3
Fig. 3

Off-axis intensity distributions of a collimated TEM 10 beam diffracted by an opaque disk and measured at N f = 7 .

Fig. 4
Fig. 4

Off-axis intensity distributions of a collimated Gaussian beam diffracted by an opaque disk and measured at N f = 7 .

Tables (3)

Tables Icon

Table 1 Gaussian Beam Diffracted by a Hard-Edge Aperture

Tables Icon

Table 2 Collimated TEM 10 Beam Diffracted by an Opaque Disk

Tables Icon

Table 3 Collimated Gaussian Beam Diffracted by an Opaque Disk

Equations (64)

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

Ψ p ( ρ , z ) = 2 π 1 w ( z ) exp [ ρ 2 w ( z ) 2 ] L p [ 2 ρ 2 w ( z ) 2 ] exp [ j k ρ 2 2 R ( z ) ] exp [ j ( 2 p + 1 ) γ ( z ) ] exp [ j k ( z z 0 ) ] ,
w ( z ) = w 0 [ 1 + ( z z 0 z R ) 2 ] 1 2 ,
R ( z ) = ( z z 0 ) [ 1 + ( z R z z 0 ) 2 ] ,
γ ( z ) = arctan ( z z 0 z R ) ,
E ( r , z ) = 2 π exp ( j k z ) j λ z exp ( j π r 2 λ z ) × 0 + τ ( ρ ) Ψ p ( ρ , 0 ) exp ( j k ρ 2 2 z ) J 0 ( 2 π λ z r ρ ) ρ d ρ ,
E ( r , z ) = C 0 + τ ( ρ ) L p [ 2 ρ 2 w ( 0 ) 2 ] exp { [ 1 w ( 0 ) 2 + j k 2 ( 1 R ( 0 ) + 1 z ) ] ρ 2 } J 0 ( b ρ ) ρ d ρ ,
C = 2 π exp ( j k z ) j λ z exp ( j π r 2 λ z ) 2 π 1 w ( 0 ) exp [ j ( 2 p + 1 ) γ ( 0 ) ] exp ( j k z 0 ) ,
b = 2 π λ z r .
J 1 ( x ) = x 0 + circ ( t ) J 0 ( t x ) t d t ,
circ ( t ) n = 0 N 1 f n ( ν 0 ) Ψ n ( t ; ν 0 ) ,
Ψ n ( t ; ν 0 ) = 2 π 1 ν 0 L n ( 2 t 2 ν 0 2 ) exp ( t 2 ν 0 2 ) ,
ν 0 = 1 N .
f n ( ν 0 ) = 2 π 0 + circ ( t ) Ψ n * ( t ; ν 0 ) d t ,
f n ( ν 0 ) = 2 π ν 0 I ( 0 , n ) ,
ϵ ̂ N ( ν 0 ) = 1 1 π n = 0 N 1 f n 2 ( ν 0 ) .
J 1 ( x ) 2 π 1 ν 0 x n = 0 N 1 [ f n ( ν 0 ) 0 + L n ( 2 t 2 ν 0 2 ) exp ( t 2 ν 0 2 ) J 0 ( x t ) t d t ] .
0 + exp ( μ t 2 ) L n ( ν t 2 ) J 0 ( t x ) t d t = ( μ ν ) n 2 μ n + 1 exp ( x 2 4 μ ) L n [ ν x 2 4 μ ( ν μ ) ] ,
J 1 ( x ) = J 0 ( x ) 1 2 μ 2 π 1 ν 0 x exp ( x 2 4 μ ) n = 0 N 1 ( 1 ) n f n ( ν 0 ) L n ( x 2 2 μ ) .
x J 0 ( x ) + J 0 ( x ) + x J 0 ( x ) = 0
p 1 , x L p ( x ) = p [ L p ( x ) L p 1 ( x ) ] ,
J 0 ( x ) 1 2 μ 2 π 1 ν 0 exp ( x 2 4 μ ) { f 0 ( ν 0 ) [ 1 + L 1 ( x 2 2 μ ) ] + n = 1 N 1 ( 1 ) n f n ( ν 0 ) [ ( n + 1 ) L n + 1 ( x 2 2 μ ) + L n ( x 2 2 μ ) n L n 1 ( x 2 2 μ ) ] } .
x k x k max , | J 0 ( x k ) J 0 N ( x k ) | ϵ .
x k max = 2.04192 N 32.7612 ,
x ¯ k max = x k max 14 = 2.04192 N 46.7612 ,
E ( r , z ) = C 0 a L p [ 2 ρ 2 w ( 0 ) 2 ] exp { [ 1 w ( 0 ) 2 + j k 2 ( 1 R ( 0 ) + 1 z ) ] ρ 2 } J 0 ( b ρ ) ρ d ρ .
E ( 0 , z ) = C 0 a L p ( α ρ 2 ) exp ( Q ρ 2 ) ρ d ρ ,
α = 2 w ( 0 ) 2 ,
Q = 1 w ( 0 ) 2 + j k 2 ( 1 R ( 0 ) + 1 z ) .
E ( 0 , z ) = C 2 I Γ ( p , 0 ) ,
E ( r , z ) = C 1 2 μ 2 π 1 ν 0 { f 0 ( ν 0 ) 0 a L p ( α ρ 2 ) [ L 0 ( β ρ 2 ) + L 1 ( β ρ 2 ) ] exp ( Q ρ 2 ) ρ d ρ + n = 1 N 1 ( 1 ) n f n ( ν 0 ) [ ( n + 1 ) 0 a L p ( α ρ 2 ) L n + 1 ( β ρ 2 ) exp ( Q ρ 2 ) ρ d ρ + 0 a L p ( α ρ 2 ) L n ( β ρ 2 ) exp ( Q ρ 2 ) ρ d ρ n 0 a L p ( α ρ 2 ) L n 1 ( β ρ 2 ) exp ( Q ρ 2 ) ρ d ρ ] } ,
α = 2 w ( 0 ) 2 ,
β = b 2 2 μ ,
Q = 1 w ( 0 ) 2 + b 2 4 μ + j k 2 ( 1 R ( 0 ) + 1 z ) .
E ( r , z ) C 2 1 2 μ 2 π 1 ν 0 { f 0 ( ν 0 ) [ I Γ ( p , 0 ) + I Γ ( p , 1 ) ] + n = 1 N 1 ( 1 ) n f n ( ν 0 ) [ ( n + 1 ) I Γ ( p , n + 1 ) + I Γ ( p , n ) n I Γ ( p , n 1 ) ] } ,
τ ( ρ ) = k ρ 2 2 f
Q = 1 w ( 0 ) 2 + b 2 4 μ + j k 2 ( 1 R ( 0 ) 1 f + 1 z ) .
τ ( ρ a ) = { 1 , ρ > a 0 , 0 ρ a } ,
E ( r , z ) = C [ 0 + L p [ 2 ρ 2 w ( 0 ) 2 ] exp { [ 1 w ( 0 ) 2 + j k 2 ( 1 R ( 0 ) + 1 z ) ] ρ 2 } J 0 ( b ρ ) ρ d ρ 0 a L p [ 2 ρ 2 w ( 0 ) 2 ] exp { [ 1 w ( 0 ) 2 + j k 2 ( 1 R ( 0 ) + 1 z ) ] ρ 2 } J 0 ( b ρ ) ρ d ρ ] .
E ( 0 , z ) = C 2 [ ( Q α ) p Q p + 1 I Γ ( p , 0 ) ] ,
E ( r , z ) C 2 [ ( Q α ) p Q p + 1 exp ( b 2 4 Q ) L p [ α b 2 4 Q ( α Q ) ] 1 2 μ 2 π 1 ν 0 { f 0 ( ν 0 ) [ I Γ ( p , 0 ) + I Γ ( p , 1 ) ] + n = 1 N 1 ( 1 ) n f n ( ν 0 ) [ ( n + 1 ) I Γ ( p , n + 1 ) + I Γ ( p , n ) n I Γ ( p , n 1 ) ] } ] ,
ϵ = i | I 1 ( i ) I 2 ( i ) | i I 1 ( i ) × 100 .
N f = ( 1 R ( 0 ) + 1 z m ) a 2 λ .
x J 1 ( x ) = 0 x u J 0 ( u ) d u .
circ ( ρ a ) = { 1 0 ρ a 0 ρ > a } ,
x J 1 ( x ) = 0 + circ ( u x ) u J 0 ( u ) d u .
circ ( u x ) n = 0 N 1 A n exp ( B n x 2 u 2 ) ,
x J 1 ( x ) n = 0 N 1 [ A n 0 + exp ( B n x 2 u 2 ) u J 0 ( u ) d u ] .
0 + exp ( p t 2 ) J 0 ( b t ) t d t = 1 2 p exp ( b 2 4 p ) ,
J 1 ( x ) = J 0 ( x ) n = 0 N 1 A n 2 B n x exp ( x 2 4 B n ) .
J 0 ( x ) n = 0 N 1 A n exp ( x 2 4 B n ) .
J 0 ( x ) 1 2 n = 0 N 1 [ A n exp ( x 2 4 B n ) + A n * exp ( x 2 4 B n * ) ] ,
I Γ ( p , q ) = 0 Γ L p ( μ x ) L q ( ν x ) exp ( R x ) d x ,
I Γ ( p , q ) = Γ 0 1 L p ( α y ) L q ( β y ) exp ( Q y ) d y = Γ I ( p , q ) ,
p 2 , p L p ( x ) = ( 2 p 1 x ) L p 1 ( x ) ( p 1 ) L p 2 ( x ) ,
I ( 0 , 0 ) = 1 Q [ 1 exp ( Q ) ] ,
I ( 0 , 1 ) = ( 1 β Q ) I ( 0 , 0 ) + β Q exp ( Q ) ,
q 2 , I ( 0 , q ) = ( 2 q 1 q β Q ) I ( 0 , q 1 ) + q 1 q ( β Q 1 ) I ( 0 , q 2 ) + β q Q exp ( Q ) L q 1 ( β ) ,
I ( 1 , 0 ) = ( 1 α Q ) I ( 0 , 0 ) + α Q exp ( Q ) ,
q 1 , I ( 1 , q ) = [ 1 α ( q + 1 ) Q ] I ( 0 , q ) + α q Q I ( 0 , q 1 ) + α Q L q ( β ) exp ( Q ) ,
p 2 , I ( p , 0 ) = ( 2 p 1 p α Q ) I ( p 1 , 0 ) + p 1 p ( α Q 1 ) I ( p 2 , 0 ) + α p Q exp ( Q ) L p 1 ( α ) ,
p 2 , q 1 , I ( p , q ) = 1 p [ T 1 ( p , q ) + T 2 ( p , q ) + T 3 ( p , q ) + α Q exp ( Q ) L p 1 ( α ) L q ( β ) ] ,
T 1 ( p , q ) = [ 2 p 1 α ( p + q ) Q ] I ( p 1 , q ) ,
T 2 ( p , q ) = ( p 1 ) ( α Q 1 ) I ( p 2 , q ) ,
T 3 ( p , q ) = α q Q I ( p 1 , q 1 ) .

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