Abstract

An improved approach called the weighted YG algorithm for the design of the diffractive phase element (DPE) that implements beam shaping in the fractional Fourier transform domain and free space is presented. Modeling designs of the DPE are carried out for several fractional orders and different param eters of the beam for optimally converting a Gaussian profile into a uniform beam. We found that our algorithm can improve the beam shaping effect, reduce the error function, and increase uniformity of light intensity.

© 2011 Optical Society of America

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  1. J. Bengtsson, “Kinoform-only Gaussian-to-rectangle beam shaper for a semiconductor laser,” Appl. Opt. 35, 3807–3814(1996).
    [CrossRef] [PubMed]
  2. J. Jia, C. Zhou, X. Sun, and L. Liu, “Superresolution laser beam shaping,” Appl. Opt. 43, 2112–2117 (2004).
    [CrossRef] [PubMed]
  3. W. Mohammed and X. Gu, “Long-period grating and its application in laser beam shaping in the 1.0 m wavelength region,” Appl. Opt. 48, 2249–2254 (2009).
    [CrossRef] [PubMed]
  4. R. Pereira, B. Weichelt, D. Liang, P. J. Morais, H. Gouveia, M. Abdou-Ahmed, A. Voss, and T. Graf, “Efficient pump beam shaping for high-power thin-disk laser systems,” Appl. Opt. 49, 5157–5162 (2010).
    [CrossRef] [PubMed]
  5. B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
    [CrossRef]
  6. Y.-H. Chang, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profile,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef]
  7. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
  8. M. D. McNeill and T. C. Poon, “Gaussian-beam profile shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4515(1994).
    [CrossRef] [PubMed]
  9. H. T. Yura and T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826–6828 (1995).
    [CrossRef] [PubMed]
  10. L. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
    [CrossRef]
  11. Z.-J. Liu, H.-F. Zhao, J.-L. Liu, M. A. Ahmad, and S. Liu, “Generation of hollow Gaussian beam by spatial filtering,” Opt. Lett. 32, 2076–2078 (2007).
    [CrossRef] [PubMed]
  12. Z.-J. Liu, J.-M. Dai, X.-G. Sun, and S.-T. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16, 19926–19933 (2008).
    [CrossRef] [PubMed]
  13. Z.-B. Tian, M. Nix, and S. H. Yam, “Laser beam shaping using a single-mode fiber abrupt taper,” Opt. Lett. 34, 229–231(2009).
    [CrossRef] [PubMed]
  14. M. Fratz, S. Sinzinger, and D. Giel, “Design and fabrication of polarization-holographic elements for laser beam shaping,” Appl. Opt. 48, 2669–2677 (2009).
    [CrossRef] [PubMed]
  15. J. Liang, R. N. Kohn Jr., M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Opt. 49, 1323–1330 (2010).
    [CrossRef] [PubMed]
  16. A. Haghighatzadeh and H. Golnabi, “Flat-top beam profile generated using a fiber-bundle prism-coupled beam shaper,” Opt. Commun. 284, 2817–2824 (2011).
    [CrossRef]
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    [CrossRef]
  18. J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2549 (1993).
    [CrossRef] [PubMed]
  19. N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
    [CrossRef]
  20. C. Zhang and A. Kar, “Diffractive optical elements for pitchfork beam shaping,” Opt. Eng. 48, 078001 (2009).
    [CrossRef]
  21. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  22. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).
  23. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  26. B.-Y. Gu, G.-Z. Yang, and B.-Z. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206(1986).
    [CrossRef] [PubMed]
  27. S.-H. Yan, “Research on the weighted Yang-Gu algorithm,” Acta Photon. Sin. 36, 530–535 (2007).
    [CrossRef]
  28. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation,” J. Opt. Soc. Am. A 10, 1875–1881 (1993).
    [CrossRef]
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    [CrossRef]
  30. S.-H. Tao and X.-C. Yuan, “Practical implementation of the phase-quantization technique in an iterative Fourier-transform algorithm,” Appl. Opt. 43, 2089–2092 (2004).
    [CrossRef] [PubMed]
  31. W. Hsu and C. Lin, “Optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,” Appl. Opt. 44, 5802–5808(2005).
    [CrossRef] [PubMed]
  32. H. Duadi and Z. Zalevsky, “Optimized iterative quantization algorithm for phase-only beam shaping masks,” Opt. Commun. 283, 951–957 (2010).
    [CrossRef]

2011

A. Haghighatzadeh and H. Golnabi, “Flat-top beam profile generated using a fiber-bundle prism-coupled beam shaper,” Opt. Commun. 284, 2817–2824 (2011).
[CrossRef]

2010

B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
[CrossRef]

H. Duadi and Z. Zalevsky, “Optimized iterative quantization algorithm for phase-only beam shaping masks,” Opt. Commun. 283, 951–957 (2010).
[CrossRef]

J. Liang, R. N. Kohn Jr., M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Opt. 49, 1323–1330 (2010).
[CrossRef] [PubMed]

R. Pereira, B. Weichelt, D. Liang, P. J. Morais, H. Gouveia, M. Abdou-Ahmed, A. Voss, and T. Graf, “Efficient pump beam shaping for high-power thin-disk laser systems,” Appl. Opt. 49, 5157–5162 (2010).
[CrossRef] [PubMed]

2009

2008

S.-G. Zhou and X.-J. Shen, “Influence of optical element misalignment of beam spread collimation optical system on Gaussian beam propagation and transformation,” J. Appl. Opt. 29, 253–256 (2008).
[CrossRef]

Z.-J. Liu, J.-M. Dai, X.-G. Sun, and S.-T. Liu, “Generation of hollow Gaussian beam by phase-only filtering,” Opt. Express 16, 19926–19933 (2008).
[CrossRef] [PubMed]

2007

2005

2004

2003

L. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

1998

1996

1995

1994

1993

1989

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

1986

1983

1982

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Abdou-Ahmed, M.

Ahmad, M. A.

Ameur, K. A.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Antonucci, L.

B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
[CrossRef]

Becker, M. F.

Bengtsson, J.

Brahim, T. M.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Chang, Y.-H.

Cordingley, J.

Dai, J.-M.

Dong, B.-Z.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Duadi, H.

H. Duadi and Z. Zalevsky, “Optimized iterative quantization algorithm for phase-only beam shaping masks,” Opt. Commun. 283, 951–957 (2010).
[CrossRef]

Eimer, S.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Ersoy, O. K.

Fienup, J. R.

Fratz, M.

Fromager, M.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Giel, D.

Golnabi, H.

A. Haghighatzadeh and H. Golnabi, “Flat-top beam profile generated using a fiber-bundle prism-coupled beam shaper,” Opt. Commun. 284, 2817–2824 (2011).
[CrossRef]

Gouveia, H.

Graf, T.

Gu, B.-Y.

Gu, X.

Gunther, C.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Haghighatzadeh, A.

A. Haghighatzadeh and H. Golnabi, “Flat-top beam profile generated using a fiber-bundle prism-coupled beam shaper,” Opt. Commun. 284, 2817–2824 (2011).
[CrossRef]

Heinzen, D. J.

Hsu, W.

Ishii, Y.

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Jia, J.

Jullien, A.

B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
[CrossRef]

Kar, A.

C. Zhang and A. Kar, “Diffractive optical elements for pitchfork beam shaping,” Opt. Eng. 48, 078001 (2009).
[CrossRef]

Kivshar, Y. S.

L. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Kohn, R. N.

Liang, D.

Liang, J.

Lin, C.

Liu, J.-L.

Liu, L.

Liu, S.

Liu, S.-T.

Liu, Z.-J.

McNeill, M. D.

Mechin, L.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Mendlovic, D.

Mercier, B.

B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
[CrossRef]

Mohammed, W.

Morais, P. J.

Murata, K.

Nix, M.

Ozaktas, H. M.

Passilly, N.

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

Pereira, R.

Poon, T. C.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Rose, T. S.

Rousseau, J. P.

B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Shadrivov, L. V.

L. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Shen, X.-J.

S.-G. Zhou and X.-J. Shen, “Influence of optical element misalignment of beam spread collimation optical system on Gaussian beam propagation and transformation,” J. Appl. Opt. 29, 253–256 (2008).
[CrossRef]

Sinzinger, S.

Streibl, N.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Sukhorukov, A. A.

L. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

Sun, X.

Sun, X.-G.

Tao, S.-H.

Tian, Z.-B.

Voss, A.

Wackerman, C. C.

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

Weichelt, B.

Yam, S. H.

Yan, S.-H.

S.-H. Yan, “Research on the weighted Yang-Gu algorithm,” Acta Photon. Sin. 36, 530–535 (2007).
[CrossRef]

Yang, G.-Z.

Yuan, X.-C.

Yura, H. T.

Zalevsky, Z.

H. Duadi and Z. Zalevsky, “Optimized iterative quantization algorithm for phase-only beam shaping masks,” Opt. Commun. 283, 951–957 (2010).
[CrossRef]

Zhang, C.

C. Zhang and A. Kar, “Diffractive optical elements for pitchfork beam shaping,” Opt. Eng. 48, 078001 (2009).
[CrossRef]

Zhang, Y.

Zhao, H.-F.

Zhou, C.

Zhou, S.-G.

S.-G. Zhou and X.-J. Shen, “Influence of optical element misalignment of beam spread collimation optical system on Gaussian beam propagation and transformation,” J. Appl. Opt. 29, 253–256 (2008).
[CrossRef]

Zhuang, J.-Y.

Acta Photon. Sin.

S.-H. Yan, “Research on the weighted Yang-Gu algorithm,” Acta Photon. Sin. 36, 530–535 (2007).
[CrossRef]

Appl. Opt.

W. Mohammed and X. Gu, “Long-period grating and its application in laser beam shaping in the 1.0 m wavelength region,” Appl. Opt. 48, 2249–2254 (2009).
[CrossRef] [PubMed]

M. Fratz, S. Sinzinger, and D. Giel, “Design and fabrication of polarization-holographic elements for laser beam shaping,” Appl. Opt. 48, 2669–2677 (2009).
[CrossRef] [PubMed]

J. Liang, R. N. Kohn Jr., M. F. Becker, and D. J. Heinzen, “High-precision laser beam shaping using a binary-amplitude spatial light modulator,” Appl. Opt. 49, 1323–1330 (2010).
[CrossRef] [PubMed]

R. Pereira, B. Weichelt, D. Liang, P. J. Morais, H. Gouveia, M. Abdou-Ahmed, A. Voss, and T. Graf, “Efficient pump beam shaping for high-power thin-disk laser systems,” Appl. Opt. 49, 5157–5162 (2010).
[CrossRef] [PubMed]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

Y.-H. Chang, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profile,” Appl. Opt. 22, 3644–3647 (1983).
[CrossRef]

B.-Y. Gu, G.-Z. Yang, and B.-Z. Dong, “General theory for performing an optical transform,” Appl. Opt. 25, 3197–3206(1986).
[CrossRef] [PubMed]

J. Cordingley, “Application of a binary diffractive optic for beam shaping in semiconductor processing by lasers,” Appl. Opt. 32, 2538–2549 (1993).
[CrossRef] [PubMed]

G.-Z. Yang, B.-Z. Dong, B.-Y. Gu, J.-Y. Zhuang, and O. K. Ersoy, “Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system:a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

M. D. McNeill and T. C. Poon, “Gaussian-beam profile shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4515(1994).
[CrossRef] [PubMed]

H. T. Yura and T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826–6828 (1995).
[CrossRef] [PubMed]

J. Bengtsson, “Kinoform-only Gaussian-to-rectangle beam shaper for a semiconductor laser,” Appl. Opt. 35, 3807–3814(1996).
[CrossRef] [PubMed]

S.-H. Tao and X.-C. Yuan, “Practical implementation of the phase-quantization technique in an iterative Fourier-transform algorithm,” Appl. Opt. 43, 2089–2092 (2004).
[CrossRef] [PubMed]

J. Jia, C. Zhou, X. Sun, and L. Liu, “Superresolution laser beam shaping,” Appl. Opt. 43, 2112–2117 (2004).
[CrossRef] [PubMed]

W. Hsu and C. Lin, “Optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm,” Appl. Opt. 44, 5802–5808(2005).
[CrossRef] [PubMed]

Appl. Phys. Lett.

L. V. Shadrivov, A. A. Sukhorukov, and Y. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003).
[CrossRef]

J. Appl. Opt.

S.-G. Zhou and X.-J. Shen, “Influence of optical element misalignment of beam spread collimation optical system on Gaussian beam propagation and transformation,” J. Appl. Opt. 29, 253–256 (2008).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

H. Duadi and Z. Zalevsky, “Optimized iterative quantization algorithm for phase-only beam shaping masks,” Opt. Commun. 283, 951–957 (2010).
[CrossRef]

N. Passilly, M. Fromager, L. Mechin, C. Gunther, S. Eimer, T. M. Brahim, and K. A. Ameur, “1-D laser beam shaping using an adjustable binary diffractive optical element,” Opt. Commun. 241, 465–473 (2004).
[CrossRef]

B. Mercier, J. P. Rousseau, A. Jullien, and L. Antonucci, “Nonlinear beam shaper for femtosecond laser pulses, from Gaussian to flat-top profile,” Opt. Commun. 283, 2900–2907 (2010).
[CrossRef]

A. Haghighatzadeh and H. Golnabi, “Flat-top beam profile generated using a fiber-bundle prism-coupled beam shaper,” Opt. Commun. 284, 2817–2824 (2011).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–306 (1980).

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, and S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).

C. Zhang and A. Kar, “Diffractive optical elements for pitchfork beam shaping,” Opt. Eng. 48, 078001 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Optik

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

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Figures (6)

Fig. 1
Fig. 1

Schematic view of a beam shaping system with two diffractive phase elements.

Fig. 2
Fig. 2

Comparison of amplitude distributions for Gaussian profile beam shaping with three different algorithms in the FRFT domain with order p = 0.6 : (a1), (b1) with the GS algorithm, (a2), (b2) with the YG algorithm, and (a3), (b3) with the weighed GS algorithm. Panels (a1)–(a3) were obtained with 10 iterations and (b1)–(b3) with 100 iterations. i1, amplitude distribution of the input Gaussian profile beam; i2, amplitude distribution of the ideal uniform beam; i3, amplitude distribution of the output beam generated by the designed DPEs.

Fig. 3
Fig. 3

Relative amplitude distributions for Gaussian beam shaping in the FRFT domain with the YG algorithm and the W-YG algorithm for three different waist radii when the fractional order is p = 0.8 . (a1)–(a3) With the YG algorithm, (b1)–(b3) with the W-YG algorithm. The corresponding waist sizes of the Gaussian profile laser beams are (a1), (b1)  ω 1 = 0.45 a 1 , (a2), (b2)  ω 1 = 0.5 a 1 , and (a3), (b3)  ω 1 = 0.875 a 1 .

Fig. 4
Fig. 4

Comparison of the convergence curves of the two algorithms with fractional orders (a)  p = 0.6 and (b)  p = 0.8 . The graphs depict the SSE error of the initial phase of ϕ 0 = 0 (solid curve) and ϕ 0 = 1.5 (dashed curve) of the YG algorithm and ϕ 0 = 0 (solid triangle curve) and ϕ 0 = 1.5 (solid dot curve) of the W-YG algorithm.

Fig. 5
Fig. 5

Comparison of phase distributions of DPE2 with order p = 0.6 of two algorithms. The phase distribution obtained (a) with the YG algorithm and (b) with the W-YG algorithm.

Fig. 6
Fig. 6

Relative amplitude distributions for Gaussian beam shaping in free space with (a) the YG algorithm and (b) the W-YG algorithm. The corresponding waist size of the Gausssian profile beam is ω 1 = 0.35 a 1 .

Equations (19)

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

U 1 i = ρ 1 i exp ( i ϕ 1 i ) i = 1 , 2 , 3 , N 1 ,
U 2 j = ρ 2 j exp ( i ϕ 2 j ) j = 1 , 2 , 3 , N 2 ,
U 2 j = i = 1 N 1 G i j U 1 i ,
ϕ 1 k = arg [ i = 1 N 2 G j k * ρ 2 j exp ( i ϕ 2 j ) j k A k j ρ 1 k exp ( i ϕ 1 k ) ] ,
ϕ 2 k = arg [ k = 1 N 1 G j k ρ 1 k exp ( i ϕ 1 k ) ] ,
ρ ¯ 2 ( n ) = ρ 20 ,
u ¯ 2 ( n ) = ρ ¯ 2 ( n ) exp ( i ϕ 2 ( n ) ) = ρ 20 u 2 ( n ) | u 2 ( n ) | ,
ρ ¯ 2 ( n ) = q ρ 20 + ( 1 q ) | u 2 ( n ) | .
u ¯ 2 ( n ) = ρ ¯ 2 ( n ) exp ( i ϕ 2 ( n ) ) = q ρ 20 u 2 ( n ) | u 2 ( n ) | + ( 1 q ) u 2 ( n ) ,
ε u = j = 1 N 2 | ρ 20 ρ 2 j | 2 .
ε u ( n ) = j = 1 N 2 | ρ 20 | u 2 j ( n ) | | 2 .
ε ¯ u ( n ) = j = 1 N 2 | ρ 20 ρ ¯ 2 j ( n ) | 2 = j = 1 N 2 | ρ 20 q ρ 20 ( 1 q ) | u 2 j ( n ) | | 2 = ( 1 q ) 2 j = 1 N 2 | ρ 20 | u 2 j ( n ) | | 2 .
SSE = [ ρ 2 ( x 2 ) | G ^ ρ 1 exp ( i ϕ ( ~ n ) ) | ] 2 / ρ 2 2 ( x 2 ) d x 2 ,
ξ = | I out I ideal | I ideal ,
U 2 ( x 2 ) = F α { U 1 ( x 1 ) } = B ( x 2 , x 1 , α ) U 1 ( x 1 ) d x 1 ,
B ( x 2 , x 1 , α ) = 1 i cot ( α ) λ F I exp { i π λ F I [ ( x 1 2 + x 2 2 ) cot ( α ) 2 x 1 x 2 csc ( α ) ] } ,
C ( α ) = 1 N 1 i = 1 N 1 A i i ,
B ( α ) = 1 N 1 ( N 1 1 ) i = 1 N 1 j i N 1 A i j .
G ( x 2 , x 1 ) = 1 j λ Z d exp ( j 2 π Z d / λ ) exp [ j π ( x 2 x 1 ) 2 / λ Z d ] .

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