Abstract

A general model for different apertures and flat-topped laser beams based on the multi-Gaussian function is developed. The general analytical expression for the propagation of a flat-topped beam through a general double-lens system with apertures is derived using the above model. Then, the propagation characteristics of the flat-topped beam through a spatial filter are investigated by using a simplified analytical expression. Based on the Fluence beam contrast and the Fill factor, the influences of a pinhole size on the propagation of the flat-topped multi-Gaussian beam (FMGB) through the spatial filter are illustrated. An analytical expression for the propagation of the FMGB through the spatial filter with a misaligned pinhole is presented, and the influences of the pinhole offset are evaluated.

© 2009 OSA

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References

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  1. H. T. Eyyuboglu, Ç. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006).
    [CrossRef] [PubMed]
  2. X. Du and D. Zhao, “Propagation of the decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems,” J. Opt. Soc. Am. A 23(3), 625–631 (2006).
    [CrossRef]
  3. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27(12), 1007–1009 (2002).
    [CrossRef]
  4. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206(4-6), 225–234 (2002).
    [CrossRef]
  5. Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44(8), 1381–1386 (2005).
    [CrossRef] [PubMed]
  6. J. Chen, “Propagation and transformation of flat-topped multi-Gaussian beams in a general nonsymmetrical apertured double-lens system,” J. Opt. Soc. Am. A 24(1), 84–92 (2007).
    [CrossRef]
  7. H. Mao and D. Zhao, “Different models for a hard-aperture function and corresponding approximate analytical propagation equation of a Gaussian beam through an apertured optical system,” J. Opt. Soc. Am. A 22(4), 647–653 (2005).
    [CrossRef]
  8. P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14(6), 1130–1135 (2005).
    [CrossRef]
  9. B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A 17(11), 2001–2004 (2000).
    [CrossRef]
  10. B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619–2718 (2001).
    [CrossRef]
  11. M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193(1-6), 73–79 (2001).
    [CrossRef]
  12. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6(4), 390–395 (2004).
    [CrossRef]
  13. Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6(12), 1061–1066 (2004).
    [CrossRef]
  14. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13(3), 201–203 (1988).
    [CrossRef] [PubMed]
  15. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
    [CrossRef]
  16. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
    [CrossRef]
  17. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18(8), 1897–1904 (2001).
    [CrossRef]
  18. H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
    [CrossRef]
  19. P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. Da Silva, B. J. Macgowan, B. M. Van Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. 37(12), 2371–2378 (1998).
    [CrossRef]
  20. A. K. Potemkin, T. V. Barmashova, A. V. Kirsanov, M. A. Martyanov, E. A. Khazanov, and A. A. Shaykin, “Spatial filters for high-peak-power multistage laser amplifiers,” Appl. Opt. 46(20), 4423–4430 (2007).
    [CrossRef] [PubMed]
  21. J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053–2057 (1978).
    [CrossRef] [PubMed]
  22. J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the national ignition facility,” Appl. Opt. 39(9), 1405–1420 (2000).
    [CrossRef]
  23. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007).
    [CrossRef] [PubMed]
  24. Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Acta. Phys. Sin-Ch. Ed. 57, 6992–6997 (2008).
  25. Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
    [CrossRef]
  26. Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
    [CrossRef] [PubMed]

2009 (2)

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

2008 (1)

H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[CrossRef]

2007 (3)

2006 (2)

2005 (3)

2004 (2)

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6(4), 390–395 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6(12), 1061–1066 (2004).
[CrossRef]

2002 (2)

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206(4-6), 225–234 (2002).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27(12), 1007–1009 (2002).
[CrossRef]

2001 (3)

A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18(8), 1897–1904 (2001).
[CrossRef]

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619–2718 (2001).
[CrossRef]

M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193(1-6), 73–79 (2001).
[CrossRef]

2000 (2)

1998 (1)

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
[CrossRef]

1988 (2)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13(3), 201–203 (1988).
[CrossRef] [PubMed]

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

1978 (1)

Arpali, Ç.

Auerbach, J. M.

Barmashova, T. V.

Baykal, Y. K.

Belafhal, A.

M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193(1-6), 73–79 (2001).
[CrossRef]

Boley, C. D.

Bowers, M. W.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

Cai, Y.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6(4), 390–395 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6(12), 1061–1066 (2004).
[CrossRef]

Caird, J. A.

Celliers, P. M.

Chen, J.

Chen, T.

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14(6), 1130–1135 (2005).
[CrossRef]

Da Silva, L. B.

De Silvestri, S.

Dixit, S. N.

Du, X.

Erbert, G. V.

Estabrook, K. G.

Eyyuboglu, H. T.

H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[CrossRef]

H. T. Eyyuboglu, Ç. Arpali, and Y. K. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14(10), 4196–4207 (2006).
[CrossRef] [PubMed]

Gao, Y. Q.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

Glaze, J. A.

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
[CrossRef]

Haynam, C. A.

Heestand, G. M.

Henesian, M. A.

Hermann, M. R.

Hunt, J. T.

Ibnchaikh, M.

M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193(1-6), 73–79 (2001).
[CrossRef]

Jancaitis, K. S.

Khazanov, E. A.

Kirsanov, A. V.

Laporta, P.

Li, Y.

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206(4-6), 225–234 (2002).
[CrossRef]

Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27(12), 1007–1009 (2002).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6(4), 390–395 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6(12), 1061–1066 (2004).
[CrossRef]

Lin, Z. Q.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

Liu, D. Z.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

Liu, X. F.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

Lü, B.

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14(6), 1130–1135 (2005).
[CrossRef]

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619–2718 (2001).
[CrossRef]

B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A 17(11), 2001–2004 (2000).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619–2718 (2001).
[CrossRef]

B. Lü and S. Luo, “General propagation equation of flatted Gaussian beams,” J. Opt. Soc. Am. A 17(11), 2001–2004 (2000).
[CrossRef]

Macgowan, B. J.

Magni, V.

Majocchi, B.

Manes, K. R.

Mao, H.

Marshall, C. D.

Martyanov, M. A.

Mehta, N. C.

Mei, Z.

Menapace, J.

Milam, D.

Moses, E.

Murray, J. E.

Murray, J. R.

Nostrand, M. C.

Orth, C. D.

Patterson, R.

Potemkin, A. K.

Renard, P. A.

Sacks, R. A.

Shaw, M. J.

Shaykin, A. A.

Simmons, W. W.

Spaeth, M.

Sutton, S. B.

Svelto, O.

Tovar, A. A.

Van Wonterghem, B. M.

Wallace, R. J.

Wegner, P. J.

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

White, R. K.

Widmayer, C. C.

Williams, W. H.

Wu, P.

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14(6), 1130–1135 (2005).
[CrossRef]

Yang, S. T.

Zhao, D.

Zhu, B. Q.

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

Appl. Opt. (7)

J. T. Hunt, J. A. Glaze, W. W. Simmons, and P. A. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053–2057 (1978).
[CrossRef] [PubMed]

P. M. Celliers, K. G. Estabrook, R. J. Wallace, J. E. Murray, L. B. Da Silva, B. J. Macgowan, B. M. Van Wonterghem, and K. R. Manes, “Spatial filter pinhole for high-energy pulsed lasers,” Appl. Opt. 37(12), 2371–2378 (1998).
[CrossRef]

J. E. Murray, D. Milam, C. D. Boley, K. G. Estabrook, and J. A. Caird, “Spatial filter pinhole development for the national ignition facility,” Appl. Opt. 39(9), 1405–1420 (2000).
[CrossRef]

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44(8), 1381–1386 (2005).
[CrossRef] [PubMed]

C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007).
[CrossRef] [PubMed]

A. K. Potemkin, T. V. Barmashova, A. V. Kirsanov, M. A. Martyanov, E. A. Khazanov, and A. A. Shaykin, “Spatial filters for high-peak-power multistage laser amplifiers,” Appl. Opt. 46(20), 4423–4430 (2007).
[CrossRef] [PubMed]

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Characteristics of beam alignment in a high power four-pass laser amplifier,” Appl. Opt. 48(8), 1591–1597 (2009).
[CrossRef] [PubMed]

Chin. Phys. (1)

P. Wu, B. Lü, and T. Chen, “Fractional Fourier transform of flat-topped multi-Gaussian beams based on the Wigner distribution function,” Chin. Phys. 14(6), 1130–1135 (2005).
[CrossRef]

Chin. Phys. B (1)

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, X. F. Liu, and Z. Q. Lin, “Algorithm of far-field centre estimation based on phase-only matched filter,” Chin. Phys. B 18(1), 215–220 (2009).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83(5), 1752–1756 (1988).
[CrossRef]

J. Mod. Opt. (1)

B. Lü and S. Luo, “Approximate propagation equations of flattened Gaussian beams passing through a paraxial ABCD system with hard-edge aperture,” J. Mod. Opt. 48, 2619–2718 (2001).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6(4), 390–395 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6(12), 1061–1066 (2004).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (3)

Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206(4-6), 225–234 (2002).
[CrossRef]

M. Ibnchaikh and A. Belafhal, “Closed-term propagation expression of flattened Gaussian beams through an apertured ABCD optical system,” Opt. Commun. 193(1-6), 73–79 (2001).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107(5-6), 335–341 (1994).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

H. T. Eyyuboglu, “Propagation of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. 40, 156–166 (2008).
[CrossRef]

Opt. Lett. (2)

Other (1)

Y. Q. Gao, B. Q. Zhu, D. Z. Liu, Z. Y. Peng, and Z. Q. Lin, “Study of mathematical model for auto-alignment in four-pass amplifier,” Acta. Phys. Sin-Ch. Ed. 57, 6992–6997 (2008).

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Figures (11)

Fig. 1
Fig. 1

Different-order Multi-Gaussian shapes when N = M, (a), (b), and (c) show the two dimensional shapes, and (d) is the profile of (a), (b), and (c).

Fig. 2
Fig. 2

Different kinds of shapes (a) circle, (b) ellipse, and (c) rectangle.

Fig. 3
Fig. 3

Double-lens system with apertures

Fig. 4
Fig. 4

Propagation of the FMGB through spatial filter at different Fresnel numbers, (a) the input FMGB, (b) the output FMGB at F=200 , (c) the output FMGB at F=4 , and (d) the output FMGB at F=0.8 .

Fig. 5
Fig. 5

(a) Normalized overlaid plots of the FMGB belonging to Fig. 4 cut along the x-axis, (b) on-axis intensity distribution of the output FMGB vs. Fresnel number.

Fig. 6
Fig. 6

Intensity distribution in the focal plane, the red one represents the distribution of an ideal square beam; others represent the distributions of the FMGB with different orders. The y-axis is logarithmic.

Fig. 7
Fig. 7

The FBCs and Fill factors of the output FMGB at different orders when the size of pinhole alter or the distance alter. (a) the FBC and Fill factor vs. size of pinhole, (b) the FBC and Fill factor vs. 1/z4 .

Fig. 8
Fig. 8

The shape of the output FMGB with different sizes of pinhole, (a) the beam envelop corresponding the min. FBC at 4.25 × DL size, (b) the beam envelop corresponding the max. FBC at 3.75 × DL size, (c) the sections of the beams at different size of pinhole.

Fig. 9
Fig. 9

The shape of the output FMGB with different pinhole offset (x0,y0) when N1=M1=8 and N3=M3=15 , (a) (0,0.1×TPD) , (b) (0,0.2×TPD) , (c) (0,0.3×TPD) , (d) (0,0.4×TPD) , (e) (0,0.5×TPD) , and (f) (0,0.6×TPD) .

Fig. 10
Fig. 10

FBC and Fill factor, (a) FBC and Fill factor vs. offset, and (b) FBC and Fill factor vs. z4, OS=0.2 means that offset is 0.2TPD , and OS=0 means that offset is 0.

Fig. 11
Fig. 11

The shape of output FMGB at difference distance with the same offset 0.2TPD  , (a) z4=2000mm , (b) z4=50000mm , (c), z4=100000mm

Equations (22)

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

MG(x)=n=NNexp[(xnωω)2]n=NNexp(n2),
MG(x,y)=n=NNm=MMexp[(xnω)2+(ymω)2ω2]n=NNm=MMexp[(n2+m2)].
ω=WN+{1ln[n=NNexp(n2)]}1/2.
E(x1,y1)=E0n1=N1N1m1=M1M1exp[(x1n1ω1)2+(y1m1ω1)2ω12]n1=N1N1m1=M1M1exp[(n12+m12)],
E(x5,y5)=exp[ik(z1+z2+z3+z4)]λ4z1z2z3z4P(x4,y4)×exp[ik2f2(x42+y42)]×P(x3,y3)×P(x2,y2) ×exp[ik2f1(x22+y22)]×E(x1,y1)exp{ik2z1[(x2x1)2+(y2y1)2]}dx1dy1 ×exp{ik2z2[(x3x2)2+(y3y2)2]}dx2dy2×exp{ik2z3[(x4x3)2+(y4y3)2]}dx3dy3 ×exp{ik2z4[(x5x4)2+(y5y4)2]}dx4dy4,
P(xj,yj)=nj=NjNjmj=MjMjexp[(xjnjωj)2+(yjmjωj)2ωj2]nj=NjNjmj=MjMjexp[(nj2+mj2)].
exp(p2x2±qx)dx=exp(q24p2)πp (Rep2>0),
E(x5 y5)=exp[ik(z1+z2+z3+z4)]λ4z1z2z3z4exp[ik2z4(x52+y52)]E0NM1×NM2×NM3×NM4πA×πB×πC×πD  ×n4=N4N4m4=M4M4n3=N3N3m3=M3M3n2=N2N2m2=M2M2n1=N1N1m1=M1M1exp[(n42+m42)(n32+m32)(n22+m22) (n12+m12)+14A[(2n1ω1)2+(2m1ω1)2]+14B[(2n2ω2ik2Az1×2n1ω1)2+(2m2ω2ik2Az1×2m1ω1)2] +14C{[2n3ω3ik2Bz2(2n2ω2ik2Az1×2n1ω1)]2+[2m3ω3ik2Bz2(2m2ω2ik2Az1×2m1ω1)]2} +14D({2n4ω4ik2Cz3[2n3ω3ik2Bz2(2n2ω2ik2Az1×2n1ω1)]ikx5z4}2 +{2m4ω4ik2Cz3[2m3ω3ik2Bz2(2m2ω2ik2Az1×2m1ω1)]iky5z4}2)],
A=1ω12ik2z1,
B=1ω22ik2z1+ik2f1ik2z2+k24z12A,
C=1ω32ik2z2ik2z3+k24z22B,
D=1ω42+ik2f2ik2z3ik2z4+k24z32C,
NMj=nj=NjNjmj=MjMjexp[(nj2+mj2)].
E(x5 y5)=exp[ik(z1+z2+z3+z4)]λ4z1z2z3z4exp[ik2z4(x52+y52)]E0NM1×NM3πA×πB×πC×πD ×n3=N3N3m3=M3M3n1=N1N1m1=M1M1exp((n32+m32)(n12+m12)+14A[(2n1ω1)2+(2m1ω1)2] +14B[(ik2Az1×2n1ω1)2+(ik2Az1×2m1ω1)2]+14C[(2n3ω3+ik2Bz2×ik2Az1×2n1ω1)2 +(2m3ω3+ik2Bz2×ik2Az1×2m1ω1)2]+14D{[ik2Cz3×(2n3ω3+ik2Bz2×ik2Az1×2n1ω1)+ikx5z4]2 +[ik2Cz3×(2m3ω3+ik2Bz2×ik2Az1×2m1ω1)+iky5z4]2}),
A=1ω12ik2z1,
B=ik2z1+k24z12A,
C=1ω32ik2f1ik2f2+k24z22B,
D=ik2z4+k24z32C,
NMj=nj=NjNjmj=MjMjexp[(nj2+mj2)].
E(x3 y3)=exp[ik(z1+z2)](iλ)2z1z2exp[ik2f1(x32+y32)]E0NM1πA×πB×n1=N1N1m1=M1M1exp{(n12+m12) +14A[(2n1ω1)2+(2m1ω1)2]+14B[(ik2Az1×2n1ω1+ikx3z2)2+(ik2Az1×2m1ω1+iky3z2)2]}.
FBC=1mni=1mj=1n(F(xi,yj)F¯F¯)2,
E(x5 y5)=exp[ik(z1+z2+z3+z4)]λ4z1z2z3z4exp[ik2z4(x52+y52)]E0NM1×NM3πA×πB×πC×πD ×n3=N3N3m3=M3M3n1=N1N1m1=M1M1exp((n3ω3+x0)2ω32(m3ω3+x0)2ω32(n12+m12)+14A[(2n1ω1)2+(2m1ω1)2] +14B[(ik2Az1×2n1ω1)2+(ik2Az1×2m1ω1)2]+14C[(2×n3ω3+x0ω32+ik2Bz2×ik2Az1×2n1ω1)2 +(2×m3ω3+y0ω32+ik2Bz2×ik2Az1×2m1ω1)2]+14D{[ik2Cz3×(2×n3ω3+x0ω32+ik2Bz2×ik2Az1×2n1ω1)+ikx5z4]2 +[ik2Cz3×(2×m3ω3+y0ω32+ik2Bz2×ik2Az1×2m1ω1)+iky5z4]2}),

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