Abstract

According to the definition of the degree of paraxiality for a monochromatic light beam, the change of the paraxiality of a Gaussian beam diffracted by a circular aperture has been investigated. When a Gaussian beam is diffracted by a circular aperture, its paraxiality decreases. The absolute and the relative changes for the degree of paraxiality are determined not only by the ratio of the aperture radius to the incident wavelength, but also by the ratio of the aperture radius to the Gaussian waist size. The change of the paraxiality of a diffracted Gaussian beam is graphically illustrated with numerical examples, and the influence of the circular aperture on the change of the paraxiality is also demonstrated.

©2009 Optical Society of America

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References

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  1. D. Zhao, H. Mao, W. Zhang, and 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]
  2. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
    [Crossref]
  3. D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt.  6, 77–83 (2004).
    [Crossref]
  4. G. Zhou, “Far-field structure of a linearly polarized plane wave diffracted by a rectangular aperture,” Opt. Laser Technol.  41, 504–508 (2009).
    [Crossref]
  5. T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am.  70, 1323–1328 (1980).
    [Crossref]
  6. K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt.  24, 1102–1106 (1985).
    [Crossref] [PubMed]
  7. P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt.  21, 522–527 (1982).
    [Crossref] [PubMed]
  8. S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt.  30, 1584–1585 (1991).
    [Crossref] [PubMed]
  9. Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt.  36, 772–778 (1997).
    [Crossref] [PubMed]
  10. B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.  28, 2440–2442 (2003).
    [Crossref] [PubMed]
  11. C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol.  39, 598–604 (2007).
    [Crossref]
  12. K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11, 1474–1480 (2003).
    [Crossref] [PubMed]
  13. Z. L. Horväth and Z. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun.  222, 51–68 (2003).
    [Crossref]
  14. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005).
    [Crossref]
  15. S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007).
    [Crossref]
  16. H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett.  5, 4–7 (2007).
  17. G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun.  281, 1929–1934 (2008).
    [Crossref]
  18. X. Du and D. Zhao, “Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23, 625–631 (2006).
    [Crossref]
  19. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  20. G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am.  69, 575–578 (1979).
    [Crossref]
  21. T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [Crossref]
  22. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt.  29, 1940–1946 (1990).
    [Crossref] [PubMed]
  23. O. E. Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett.  33, 1360–1362 (2008).
    [Crossref] [PubMed]
  24. O. E. Gawhary and S. Severini, “Dependence of the degree of paraxiality on field correlations,” Opt. Lett.  33, 1866–1868 (2008).
    [Crossref] [PubMed]
  25. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980).

2009 (1)

G. Zhou, “Far-field structure of a linearly polarized plane wave diffracted by a rectangular aperture,” Opt. Laser Technol.  41, 504–508 (2009).
[Crossref]

2008 (3)

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun.  281, 1929–1934 (2008).
[Crossref]

O. E. Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett.  33, 1360–1362 (2008).
[Crossref] [PubMed]

O. E. Gawhary and S. Severini, “Dependence of the degree of paraxiality on field correlations,” Opt. Lett.  33, 1866–1868 (2008).
[Crossref] [PubMed]

2007 (3)

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol.  39, 598–604 (2007).
[Crossref]

S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007).
[Crossref]

H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett.  5, 4–7 (2007).

2006 (1)

2005 (1)

2004 (2)

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt.  6, 77–83 (2004).
[Crossref]

2003 (4)

D. Zhao, H. Mao, W. Zhang, and 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]

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.  28, 2440–2442 (2003).
[Crossref] [PubMed]

Z. L. Horväth and Z. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun.  222, 51–68 (2003).
[Crossref]

K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11, 1474–1480 (2003).
[Crossref] [PubMed]

1997 (1)

Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt.  36, 772–778 (1997).
[Crossref] [PubMed]

1991 (1)

S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt.  30, 1584–1585 (1991).
[Crossref] [PubMed]

1990 (1)

S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt.  29, 1940–1946 (1990).
[Crossref] [PubMed]

1985 (2)

K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt.  24, 1102–1106 (1985).
[Crossref] [PubMed]

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[Crossref]

1982 (1)

P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt.  21, 522–527 (1982).
[Crossref] [PubMed]

1980 (1)

T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am.  70, 1323–1328 (1980).
[Crossref]

1979 (1)

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am.  69, 575–578 (1979).
[Crossref]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am.  69, 575–578 (1979).
[Crossref]

Alda, J.

S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt.  30, 1584–1585 (1991).
[Crossref] [PubMed]

Belland, P.

P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt.  21, 522–527 (1982).
[Crossref] [PubMed]

Bernabeu, E.

S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt.  30, 1584–1585 (1991).
[Crossref] [PubMed]

Bor, Z.

Z. L. Horväth and Z. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun.  222, 51–68 (2003).
[Crossref]

Cheng, C.

S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007).
[Crossref]

Chu, X.

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun.  281, 1929–1934 (2008).
[Crossref]

Crenn, J. P.

P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt.  21, 522–527 (1982).
[Crossref] [PubMed]

Du, X.

Duan, K.

K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11, 1474–1480 (2003).
[Crossref] [PubMed]

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.  28, 2440–2442 (2003).
[Crossref] [PubMed]

Fukumitsu, O.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[Crossref]

T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am.  70, 1323–1328 (1980).
[Crossref]

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Dependence of the degree of paraxiality on field correlations,” Opt. Lett.  33, 1866–1868 (2008).
[Crossref] [PubMed]

O. E. Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett.  33, 1360–1362 (2008).
[Crossref] [PubMed]

Gradshteyn, S.

S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980).

Horväth, Z. L.

Z. L. Horväth and Z. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun.  222, 51–68 (2003).
[Crossref]

Huang, X.

H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett.  5, 4–7 (2007).

Jiang, Z.

Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt.  36, 772–778 (1997).
[Crossref] [PubMed]

Jing, F.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

Kakeya, M.

T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am.  70, 1323–1328 (1980).
[Crossref]

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Liang, R.

H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett.  5, 4–7 (2007).

Liu, H.

H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett.  5, 4–7 (2007).

D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt.  6, 77–83 (2004).
[Crossref]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

Liu, Z.

Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt.  36, 772–778 (1997).
[Crossref] [PubMed]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Lu, Q.

Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt.  36, 772–778 (1997).
[Crossref] [PubMed]

Lü, B.

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.  28, 2440–2442 (2003).
[Crossref] [PubMed]

K. Duan and B. Lü, “Nonparaxial analysis of far-field properties of Gaussian beams diffracted at a circular aperture,” Opt. Express 11, 1474–1480 (2003).
[Crossref] [PubMed]

Mao, H.

H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005).
[Crossref]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt.  6, 77–83 (2004).
[Crossref]

D. Zhao, H. Mao, W. Zhang, and 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]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Mizokami, H.

K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt.  24, 1102–1106 (1985).
[Crossref] [PubMed]

Nemoto, S.

S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt.  29, 1940–1946 (1990).
[Crossref] [PubMed]

Pattanayak, D. N.

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am.  69, 575–578 (1979).
[Crossref]

Ryzhik, I. M.

S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980).

Saga, N.

K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt.  24, 1102–1106 (1985).
[Crossref] [PubMed]

Severini, S.

O. E. Gawhary and S. Severini, “Dependence of the degree of paraxiality on field correlations,” Opt. Lett.  33, 1866–1868 (2008).
[Crossref] [PubMed]

O. E. Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett.  33, 1360–1362 (2008).
[Crossref] [PubMed]

Takenaka, T.

T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation for light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[Crossref]

T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am.  70, 1323–1328 (1980).
[Crossref]

Tanaka, K.

K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt.  24, 1102–1106 (1985).
[Crossref] [PubMed]

Teng, S.

S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007).
[Crossref]

Wang, L.

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol.  39, 598–604 (2007).
[Crossref]

Wang, S.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

D. Zhao, H. Mao, W. Zhang, and 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]

S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt.  30, 1584–1585 (1991).
[Crossref] [PubMed]

Wei, X.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

Yokota, M.

Zhang, W.

D. Zhao, H. Mao, W. Zhang, and 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]

Zhang, Y.

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol.  39, 598–604 (2007).
[Crossref]

Zhao, D.

X. Du and D. Zhao, “Propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23, 625–631 (2006).
[Crossref]

H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equations of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22, 647–653 (2005).
[Crossref]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt.  6, 77–83 (2004).
[Crossref]

D. Zhao, H. Mao, W. Zhang, and 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]

Zheng, C.

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol.  39, 598–604 (2007).
[Crossref]

Zheng, J.

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun.  281, 1929–1934 (2008).
[Crossref]

Zhou, G.

G. Zhou, “Far-field structure of a linearly polarized plane wave diffracted by a rectangular aperture,” Opt. Laser Technol.  41, 504–508 (2009).
[Crossref]

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun.  281, 1929–1934 (2008).
[Crossref]

Zhou, T.

S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007).
[Crossref]

Appl. Opt (5)

K. Tanaka, N. Saga, and H. Mizokami, “Field spread of a diffracted Gaussian beam through a circular aperture,” Appl. Opt.  24, 1102–1106 (1985).
[Crossref] [PubMed]

P. Belland and J. P. Crenn, “Changes in the characteristics of a Gaussian beam weakly diffracted by a circular aperture,” Appl. Opt.  21, 522–527 (1982).
[Crossref] [PubMed]

S. Wang, E. Bernabeu, and J. Alda, “ABCD matrix for weakly apertured Gaussian beams in the far field,” Appl. Opt.  30, 1584–1585 (1991).
[Crossref] [PubMed]

Z. Jiang, Q. Lu, and Z. Liu, “Relative phase shifts of apertured Gaussian beams and transformation of a Gaussian beam through a phase aperture,” Appl. Opt.  36, 772–778 (1997).
[Crossref] [PubMed]

S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt.  29, 1940–1946 (1990).
[Crossref] [PubMed]

Chin. Opt. Lett (1)

H. Liu, R. Liang, and X. Huang, “Study on the approximative formula for the far-field of a Gaussian beam under circular aperture diffraction and its divergence,” Chin. Opt. Lett.  5, 4–7 (2007).

J. Opt. A: Pure Appl. Opt (1)

D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A: Pure Appl. Opt.  6, 77–83 (2004).
[Crossref]

J. Opt. Soc. Am (2)

T. Takenaka, M. Kakeya, and O. Fukumitsu, “Asymptotic representation of the boundary diffraction wave for a Gaussian beam incident on a circular aperture,” J. Opt. Soc. Am.  70, 1323–1328 (1980).
[Crossref]

G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am.  69, 575–578 (1979).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun (4)

G. Zhou, X. Chu, and J. Zheng, “Analytical structure of an apertured vector Gaussian beam in the far field,” Opt. Commun.  281, 1929–1934 (2008).
[Crossref]

Z. L. Horväth and Z. Bor, “Focusing of truncated Gaussian beams,” Opt. Commun.  222, 51–68 (2003).
[Crossref]

D. Zhao, H. Mao, W. Zhang, and 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]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun.  236, 225–235 (2004).
[Crossref]

Opt. Express (1)

Opt. Laser Technol (2)

C. Zheng, Y. Zhang, and L. Wang, “Propagation of vectorial Gaussian beams behind a circular aperture,” Opt. Laser Technol.  39, 598–604 (2007).
[Crossref]

G. Zhou, “Far-field structure of a linearly polarized plane wave diffracted by a rectangular aperture,” Opt. Laser Technol.  41, 504–508 (2009).
[Crossref]

Opt. Lett (3)

B. Lü and K. Duan, “Nonparaxial propagation of vectorial Gaussian beams diffracted at a circular aperture,” Opt. Lett.  28, 2440–2442 (2003).
[Crossref] [PubMed]

O. E. Gawhary and S. Severini, “Degree of paraxiality for monochromatic light beams,” Opt. Lett.  33, 1360–1362 (2008).
[Crossref] [PubMed]

O. E. Gawhary and S. Severini, “Dependence of the degree of paraxiality on field correlations,” Opt. Lett.  33, 1866–1868 (2008).
[Crossref] [PubMed]

Optik (1)

S. Teng, T. Zhou, and C. Cheng, “Fresnel diffraction of truncated Gaussian beam,” Optik 118, 435–439 (2007).
[Crossref]

Phys. Rev. A (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[Crossref]

Other (1)

S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, New York, 1980).

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Figures (2)

Fig. 1.
Fig. 1. The degree of paraxiality, the absolute change for the degree of paraxiality, and the relative change for the degree of paraxiality of different Gaussian beams diffracted by a circular aperture as a function of the ratio R/λ, respectively.
Fig. 2.
Fig. 2. The degree of paraxiality, the absolute change for the degree of paraxiality, and the relative change for the degree of paraxiality of different Gaussian beams diffracted by a circular aperture as a function of the ratio R/w 0, respectively.

Equations (19)

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P=p2+q2<1AE(0)(p,q)2(1p2q2)1/2dpdqAE(0)(p,q)2dpdq,
{Ex(ρ0,0)=exp(ρ02w02)exp(iωt)Ey(ρ0,0)=0,
P0=1iπ2fexp (12f2) erf (1i2f) ,
{Ex(ρ0,0)=exp(ρ02w02)circ(ζ)exp(iωt)Ey(ρ0,0)=0,
circ(ζ)={10ζ<10ζ1 .
E(ρ,z)=exp(iωt) AE(0) (p,q) exp [ik(px+qy+γz)] dpdq ,
AE(0)(p.q)=iλ2 Ex (ρ0,0)exp[ik(px0+qy0)] d x0 d y0 =πw02λ2exp(β)n=1 (2β)nJn(αb)(αb)n i ,
xn+1Jn(x)=ddx[xn+1Jn+1(x)] .
P1=01(n=1(2β)2nJn2(αb)(αb)2n+2n=1l=1n1(2β)n+lJn(αb)Jl(αb)(αb)n+l)(1b2)1/2bdb0(n=1(2β)2nJn2(αb)(αb)2n+2n=1l=1n1(2β)n+lJn(αb)Jl(αb)(αb)n+l)bdb.
0Jn(αb)Jl(αb)b(n+l1)db=αn+l22n+l2(l1)!(n1)!(n+l2) ,
P1=α2201(n=1(2β)2nJn2(αb)(αb)2n+2n=1l=1n1(2β)n+lJn(αb)Jl(αb)(αb)n+l)(1b2)1/2bdbn=1β2n[(n1)!]2(2n1)+n=1l=1n12βn+l(n1)!(l1)!(n+l1).
Jn(αb)Jl(αb)=1l! m=0 (1)mF(m,nm;l+1;1)m!(n+m)! (αb2)2m+n+l ,
Jn2(αb)=m=0 (1)mm!(2n+m)!(n+m)! (αb2)2n+2m ,
b=sinθ, 0θπ/2,
γ=1b2=1sin2θ=cosθ.
0π/2sin2m+1θdθ=22m(m!)2(2m+1)!
P1=n=1m=0(1)m2m+n(m+1)!α2mβ2n(2n+m)!(n+m)!(2m+3)!+n=1l=1n1m=02(1)m(m+1)!F(m,nm;l+1;1)α2mβn+ll!(n+m)!(2m+3)!1α2(n=1β2n[(n1)!]2(2n1)+n=1l=1n12βn+l(n1)!(l1)!(n+l1))
ΔP=P0P1.
ΔPP0=P0P1P0 ×100%.

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