Abstract

We present a beam shaping technique in controlling the complex amplitude of an optical beam. The constraint on the amplitude of the output beam in the Gerchberg-Saxton algorithm is replaced with constraints both on the amplitude and phase of the output beam in the proposed method. The total areas of the constrained regions and free regions on the complex amplitude of the output beam in the proposed method are maintained. An output beam with arbitrary complex amplitude can be realized with the proposed method. The computing result from the proposed method is a phase-only distribution, which can be fabricated as diffractive optical element for higher diffraction efficiency. Both simulations and experiments are present and the effectiveness of the proposed method is verified.

© 2015 Optical Society of America

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2014 (2)

2013 (2)

2010 (1)

2008 (2)

2007 (1)

2004 (1)

2003 (1)

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

2002 (1)

1999 (2)

1996 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Abramochkin, E.

Alieva, T.

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref] [PubMed]

Arrizón, V.

Bernet, S.

Campos, J.

Carrada, R.

Castro, I.

Chavel, P.

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

Chen, Y.

Cottrell, D. M.

Davis, J. A.

Fienup, J. R.

Gahagan, K. T.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

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 (Stuttg.) 35(2), 237–246 (1972).

González, L. A.

Grier, D. G.

S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18(7), 6988–6993 (2010).
[Crossref] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref] [PubMed]

Hooker, S. M.

Jesacher, A.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Lee, S.-H.

Liu, F.

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

Liu, J. S.

Liu, L. Z.

Lloyd, D. T.

Maurer, C.

Moreno, I.

Mu, G.

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

O’Keeffe, K.

Ritsch-Marte, M.

Rodrigo, J. A.

Roichman, Y.

S.-H. Lee, Y. Roichman, and D. G. Grier, “Optical solenoid beams,” Opt. Express 18(7), 6988–6993 (2010).
[Crossref] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref] [PubMed]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref] [PubMed]

Ruiz, U.

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 (Stuttg.) 35(2), 237–246 (1972).

Schwaighofer, A.

Song, H.

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref] [PubMed]

Swartzlander, G. A.

Taghizadeh, M. R.

Tao, S. H.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Wang, Z.

Yang, X.

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

Yuan, X. C.

Yuan, Z. Z.

Z. Z. Yuan and S. H. Tao, “Generation of phase-gradient optical beams with an iterative algorithm,” J. Opt. 16(10), 105701 (2014).
[Crossref]

Yzuel, M. J.

Zhai, H.

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

Zhou, G.

Appl. Opt. (4)

J. Opt. (1)

Z. Z. Yuan and S. H. Tao, “Generation of phase-gradient optical beams with an iterative algorithm,” J. Opt. 16(10), 105701 (2014).
[Crossref]

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

Opt. Commun. (1)

H. Zhai, F. Liu, X. Yang, G. Mu, and P. Chavel, “Improving binary images reconstructed from kinoforms by amplitude adjustment,” Opt. Commun. 219(1–6), 81–85 (2003).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Phys. Rev. Lett. (1)

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, “Optical forces arising from phase gradients,” Phys. Rev. Lett. 100(1), 013602 (2008).
[Crossref] [PubMed]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220(4598), 671–680 (1983).
[Crossref] [PubMed]

Other (2)

www.lighttrans.com .

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

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Figures (10)

Fig. 1
Fig. 1

Schematic of constraints and freedoms in the proposed method. ‘a’ and ‘p’ represent the domains of the amplitude and phase of the input plane, respectively, and ‘A’ and ‘P’ represent the domains of the amplitude and phase of the output plane, respectively. The blank regions represent the constrained domains, and the gridding regions represent the free domains.

Fig. 2
Fig. 2

A flow chart of the proposed method.

Fig. 3
Fig. 3

Beam shaping of a line beam with gradient phase. The target (a) intensity and (b) phase profiles of the output beam, the reconstructed (c) intensity and (d) phase profiles of the output beam by the Fourier transforms, and the reconstructed (e) intensity and (f) phase profiles of the output beam by the APWS method, respectively.

Fig. 4
Fig. 4

Plots of (a) intensity and (b) phase distributions of the reconstructed line beam by the Fourier transforms, respectively. Plots of (c) intensity and (d) phase distributions of the reconstructed line beam by the APWS method, respectively.

Fig. 5
Fig. 5

Phase distributions of the DOEs computed by (a) the Fourier transforms and (b) the APWS method, respectively.

Fig. 6
Fig. 6

A schematic of the experimental setup for beam shaping.

Fig. 7
Fig. 7

CCD captured intensity distributions of the Fourier transform reconstructed line beams with phase gradients from (a) 0 to 2 π , (b) 0 to 20 π , (c) 2 π to 0, and (d) 20 π to 0, respectively.

Fig. 8
Fig. 8

Beam shaping of a vortex beam. (a) The target intensity distribution and (b) phase distribution of the output beam obtained by the Fourier transforms. (c) The reconstructed intensity distribution and (d) the phase distribution of the DOE.

Fig. 9
Fig. 9

Beam shaping of a vortex beam. (a) The target intensity distribution and (b) phase distribution of the output beam obtained by the APWS method. (c) The reconstructed intensity distribution and (d) phase distribution of the DOE.

Fig. 10
Fig. 10

(a) The reconstructed vortex beam by the computed DOE and (b) the CCD captured video frames showing that two particles trapped by the reconstructed beam are in rotation.

Equations (3)

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CAmp=Ampexp(iPh), Amp=Am p t A+Am p c (IA), Ph=P h t P+P h c (IP),
{ T=FT( ), T 1 =iFT( ) or, T=iFT[FT( )h], T 1 =iFT[FT( )ih] , h=exp[i2πz 1 λ 2 ( n x L ) 2 ( n y L ) 2 ], ih=exp[i2πz 1 λ 2 ( n x L ) 2 ( n y L ) 2 ],
η= kx=1,ky=1 Nx,Ny | W(kx,ky)Am p c (kx,ky) | 2 kx=1,ky=1 Nx,Ny | Am p c (kx,ky) | 2 , ε= kx=1,ky=1 Nx,Ny | W(kx,ky)[P h c (kx,ky)P h t (kx,ky)] | kx=1,ky=1 Nx,Ny | W(kx,ky)P h t (kx,ky) |

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