Abstract

On the basis of the truncated second-order moments method in the cylindrical coordinate systems and the expansion of the hard-edged aperture function into a finite sum of complex Gaussian functions, an approximate method used to calculate the generalized beam propagation factor (M2 factor) is proposed. The approximate analytical expressions of the generalized M2 factor for rotationally symmetric hard-edged diffracted flattened Gaussian beams defined by Gori [ Opt. Commun. 107, 335 ( 1994)] and Li [ Opt. Lett. 27, 1007 ( 2002)] are derived, respectively; we show that it depends on the beam order N and the beam truncation parameter δ. Some typical numerical examples are given to illustrate its applications that we compare by using the obtained analytical method and the numerical integration method.

© 2005 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  2. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  3. R. Martinez-Herrero, P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef] [PubMed]
  4. R. Martinez-Herrero, P. M. Mejias, M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
    [CrossRef] [PubMed]
  5. C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
    [CrossRef]
  6. P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
    [CrossRef]
  7. M. Scholl, S. Müffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
    [CrossRef]
  8. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  9. B. Lü, S. Luo, “Generalized M2 factor of hard-edged diffracted flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 2098–2102 (2001).
    [CrossRef]
  10. Y. Li, “Light beams with flat-toped profiles,” Opt. Lett. 27, 1007–1009 (2002).
    [CrossRef]
  11. J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  12. D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
    [CrossRef]
  13. D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian beams,” J. Opt. 6, 77–83 (2004).
  14. A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

2004 (1)

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian beams,” J. Opt. 6, 77–83 (2004).

2003 (1)

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

2002 (1)

2001 (1)

1996 (1)

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

1995 (1)

1994 (2)

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

1993 (1)

1988 (1)

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

Arias, M.

Belanger, P.-A.

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Breazeale, M. A.

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

Champagne, Y.

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Li, Y.

Liu, H.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian beams,” J. Opt. 6, 77–83 (2004).

Lü, B.

Luo, S.

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Mao, H.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian beams,” J. Opt. 6, 77–83 (2004).

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Martinez-Herrero, R.

Mejias, P. M.

Müffer, S.

M. Scholl, S. Müffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Pare, C.

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Post, O.

M. Scholl, S. Müffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

Scholl, M.

M. Scholl, S. Müffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

Wang, S.

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Wen, J. J.

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

Zhang, W.

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

Zhao, D.

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian beams,” J. Opt. 6, 77–83 (2004).

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

J. Acoust. Soc. Am. (1)

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

J. Opt. (1)

D. Zhao, H. Mao, H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian beams,” J. Opt. 6, 77–83 (2004).

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

Opt. Commun. (4)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

D. Zhao, H. Mao, W. Zhang, S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned ABCD optical system,” Opt. Commun. 224, 5–12 (2003).
[CrossRef]

C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Opt. Lett. (3)

Other (4)

M. Scholl, S. Müffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

A. Erdelyi, W. Magnus, F. Oberhettinger, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

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Figures (5)

Fig. 1
Fig. 1

Intensity distributions of two kinds of flattened Gaussian beam with different beam orders at the input plane: (a) flattened Gaussian beams defined by Gori, (b) flattened Gaussian beams defined by Li.

Fig. 2
Fig. 2

Generalized M2 factor of truncated flattened Gaussian beams defined by Gori as a function of beam truncation parameter δ for different beam order N. The solid curves represent the case of the approximate analytical formula, and the dashed curves denote the case of the numerical integral calculation method.

Fig. 3
Fig. 3

Generalized M2 factor of truncated flattened Gaussian beams defined by Gori as a function of beam order N for different beam truncation parameter δ. Solid curves, approximate analytical formula; dashed curves, numerical integral calculation method.

Fig. 4
Fig. 4

Generalized M2 factor of truncated flattened Gaussian beams defined by Li as a function of beam truncation parameter δ for different beam order N. Solid curves, approximate analytical formula; dashed curves, numerical integral calculation method.

Fig. 5
Fig. 5

Generalized M2 factor of truncated flattened Gaussian beams defined by Li as a function of beam order N for different beam truncation parameter δ. Solid curves, approximate analytical formula; dashed curves, numerical integral calculation method.

Equations (34)

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r 2 = 1 I 0 0 a r 2 E ( r , 0 ) 2 r d r ,
p 2 = 1 k 2 I 0 0 a E ( r , 0 ) 2 r d r + 16 3 k 2 I 0 E ( a , 0 ) 2 ,
r p = 1 2 i k I 0 0 a { r [ E ( r , 0 ) ] * E ( r , 0 ) - r E ( r , 0 ) E * ( r , 0 ) } r d r ,
I 0 = 0 a E ( r , 0 ) 2 r d r
M G 2 = k ( r 2 p 2 - r p 2 ) 1 / 2 .
A p ( r ) = { 1 r a 0 r > a .
A p ( r ) = h = 1 M A h exp ( - B h a 2 r 2 ) ,
A p ( r ) = Re [ h = 1 M A h exp ( - B h a 2 r 2 ) ] .
r 2 = 1 I 0 Re [ h = 1 M A h 0 exp ( - B h a 2 r 2 ) × r 2 E ( r , 0 ) 2 r d r ] ,
p 2 = 1 k 2 I 0 Re [ h = 1 M A h 0 exp ( - B h a 2 r 2 ) E ( r , 0 ) 2 × r d r ] + 16 3 k 2 I 0 E ( a , 0 ) 2 ,
r p = 1 2 i k I 0 Re ( h = 1 M A h 0 exp ( - B h a 2 r 2 ) × { r [ E ( r , 0 ) ] * E ( r , 0 ) - r E ( r , 0 ) E * ( r , 0 ) } r d r ) ,
I 0 = Re [ h = 1 M A h 0 exp ( - B h a 2 r 2 ) E ( r , 0 ) 2 r d r ] .
E ( r , 0 ) = exp [ - ( N + 1 ) r 2 w 0 2 ] n = 0 N 1 n ! [ ( N + 1 ) r 2 w 0 2 ] n ,
E ( r , 0 ) = n = 0 N c n L n [ 2 ( N + 1 ) r 2 w 0 2 ] exp [ - ( N + 1 ) r 2 w 0 2 ] ,
c n = ( - 1 ) n m = n N ( m n ) 1 2 m .
0 exp ( - λ x 2 ) x 2 n + 1 d x = n ! 2 λ n + 1 ,
I 0 = w 0 2 N + 1 Re [ h = 1 M n 1 = 0 N n 2 = 0 N A h ( n 1 + n 2 ) ! 2 λ n 1 + n 2 + 1 n 1 ! n 2 ! ] ,
r 2 = w 0 4 I 0 ( N + 1 ) 2 Re [ h = 1 M n 1 = 0 N n 2 = 0 N A h ( n 1 + n 2 + 1 ) ! 2 λ n 1 + n 2 + 2 n 1 ! n 2 ! ] ,
p 2 = 4 k 2 I 0 Re { h = 1 M n 1 = 0 N n 2 = 0 N A h ( n 1 + n 2 + 1 ) ! 2 λ n 1 + n 2 + 2 n 1 ! n 2 ! × [ 1 - n 1 λ n 1 + n 2 + 1 - n 2 λ n 1 + n 2 + 1 + n 1 n 2 λ 2 ( n 1 + n 2 + 1 ) ( n 1 + n 2 ) ] } + 16 3 k 2 I 0 × exp [ - 2 ( N + 1 ) δ 2 ] n 1 = 0 N n 2 = 0 N 1 n 1 ! n 2 ! × [ ( N + 1 ) δ 2 ] n 1 + n 2 ,
r p = 0 ,
λ = 2 + B h ( N + 1 ) δ 2 .
δ = a w 0
M G 2 = { Re [ h = 1 M n 1 = 0 N n 2 = 0 N A h ( n 1 + n 2 + 1 ) ! λ n 1 + n 2 + 2 n 1 ! n 2 ! ] } 1 / 2 × ( Re { h = 1 M n 1 = 0 N n 2 = 0 N A h ( n 1 + n 2 + 1 ) ! λ n 1 + n 2 + 2 n 1 ! n 2 ! × [ 1 - n 1 λ n 1 + n 2 + 1 - n 2 λ n 1 + n 2 + 1 + n 1 n 2 λ 2 ( n 1 + n 2 + 1 ) ( n 1 + n 2 ) ] } + 3 8 exp [ - 2 ( N + 1 ) δ 2 ] n 1 = 0 N n 2 = 0 N 1 n 1 ! n 2 ! × [ ( N + 1 ) δ 2 ] n 1 + n 2 ) 1 / 2 × { Re [ h = 1 M n 1 = 0 N n 2 = 0 N A h ( n 1 + n 2 ) ! 2 λ n 1 + n 2 + 1 n 1 ! n 2 ! ] } - 1 .
M G 2 = δ M 2 = [ n 1 = 0 N n 2 = 0 N ( n 1 + n 2 + 1 ) ! 2 n 1 + n 2 + 2 n 1 ! n 2 ! ] 1 / 2 × { n 1 = 0 N n 2 = 0 N ( n 1 + n 2 + 1 ) ! 2 n 1 + n 2 + 2 n 1 ! n 2 ! [ 1 - 2 n 1 n 1 + n 2 + 1 - 2 n 2 n 1 + n 2 + 1 + 4 n 1 n 2 ( n 1 + n 2 + 1 ) ( n 1 + n 2 ) ] } 1 / 2 × [ n 1 = 0 N n 2 = 0 N ( n 1 + n 2 ) ! 2 n 1 + n 2 + 2 n 1 ! n 2 ! ] - 1 ,
M G 2 = ( Re { ( h = 1 M A h λ 2 ) [ h = 1 M A h λ 2 + 8 3 exp ( - 2 δ 2 ) ] } ) 1 / 2 Re ( h = 1 N A h 2 λ ) .
M G 2 = δ M 2 = 1 ,
E ( r , 0 ) = n = 1 N ( - 1 ) n - 1 N ( N n ) exp ( - n r 2 w 0 2 ) ,
M G 2 = [ Re ( h = 1 M n 1 = 1 N n 2 = 1 N A h f ( n 1 , n 2 ) λ 2 ) x 2 ] 1 / 2 × [ Re ( h = 1 M n 1 = 1 N n 2 = 1 N A h f ( n 1 , n 2 ) λ 2 ) + 8 3 n 1 N n 2 N f ( n 1 , n 2 ) exp [ - ( n 1 + n 2 ) δ 2 ] ] 1 / 2 × [ Re ( h = 1 M n 1 = 1 N n 2 = 1 N A h f ( n 1 , n 2 ) 2 λ ) ] - 1 ,
f ( n 1 , n 2 ) = ( - 1 ) n 1 + n 2 - 2 N 2 ( N n 1 ) ( N n 2 ) ,
λ = n 1 + n 2 + B h δ 2 .
δ = a w 0
M G 2 = δ M 2 = [ n 1 = 1 N n 2 = 1 N f ( n 1 , n 2 ) ( n 1 + n 2 ) 2 ] 1 / 2 [ n 1 = 1 N n 2 = 1 N n 1 n 2 f ( n 1 , n 2 ) ( n 1 + n 2 ) 2 ] 1 / 2 n 1 = 1 N n 2 = 1 N f ( n 1 , n 2 ) 2 ( n 1 + n 2 ) .
E ( r , 0 ) = { 1 r a 0 r > a .
M G 2 = 4 3 / 3 ,

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