Abstract

Iterative algorithms based on Fourier transform are used for the design of diffractive optical elements (DOEs), which produce a given intensity distribution, usually at the far field. For the near field, these algorithms can also be used by changing the Fourier transform for the Fresnel transform. However, when the distance between the DOE and the observation plane is short, the results obtained with this modification are not always valid. In the present work, we develop a technique for obtaining the desired intensity distribution in the near field using two DOEs in tandem. We have designed an algorithm based on the standard Gerchberg–Saxton algorithm to determine the modulation of the two DOEs. The best results are obtained when the first DOE modulates the amplitude and the second DOE modulates the phase.

© 2011 Optical Society of America

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References

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2009 (2)

2008 (2)

2007 (1)

2006 (2)

J. S. Liu, M. J. Thomson, and M. R. Taghizadeh, “Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping,” J. Mod. Opt. 53, 461–471 (2006).
[CrossRef]

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
[CrossRef] [PubMed]

2005 (2)

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

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]

2004 (2)

2002 (1)

1997 (1)

H.P.Herzig, Micro-Optics, Elements, Systems and Applications, 1st ed. (Taylor & Francis, 1997).

1990 (1)

1989 (1)

1988 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–305 (1980).

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–248 (1972).

Alieva, T.

Bernabeu, E.

Bernet, S.

Bryngdahl, O.

Calvo, M.

Cambril, E.

Collin, S.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–305 (1980).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–248 (1972).

Guérineau, N.

Haidar, R.

Hsu, W.

Jesacher, A.

Kim, H.

Lee, B.

Lin, C.

Liu, J.

Liu, J. S.

J. S. Liu, M. J. Thomson, and M. R. Taghizadeh, “Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping,” J. Mod. Opt. 53, 461–471 (2006).
[CrossRef]

Maurer, C.

Pedrini, G.

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

Pelouard, J.-L.

Primot, J.

Ritsch-Marte, M.

Rodrigo, J.

Sanchez-Brea, L. M.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–248 (1972).

Schwaighofer, A.

Shen, F.

Shen, J. L.

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

Taghizadeh, M.

Taghizadeh, M. R.

J. S. Liu, M. J. Thomson, and M. R. Taghizadeh, “Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping,” J. Mod. Opt. 53, 461–471 (2006).
[CrossRef]

Thomson, M.

Thomson, M. J.

J. S. Liu, M. J. Thomson, and M. R. Taghizadeh, “Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping,” J. Mod. Opt. 53, 461–471 (2006).
[CrossRef]

Torcal-Milla, F. J.

Vincent, G.

Wang, A.

Wyrowski, F.

Yang, B.

Zhang, C. L.

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

Zhang, Y.

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

Zheng, D. X.

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

Appl. Opt. (5)

J. Mod. Opt. (1)

J. S. Liu, M. J. Thomson, and M. R. Taghizadeh, “Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping,” J. Mod. Opt. 53, 461–471 (2006).
[CrossRef]

J. Opt. Soc. Am. A (4)

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

Opt. Commun. (1)

D. X. Zheng, Y. Zhang, J. L. Shen, C. L. Zhang, and G. Pedrini, “Wave field reconstruction from a hologram sequence,” Opt. Commun. 249, 73–77 (2005).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holography,” Opt. Eng. 19, 297–305 (1980).

Opt. Express (2)

Opt. Lett. (1)

Optik (Jena) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–248 (1972).

Other (2)

H.P.Herzig, Micro-Optics, Elements, Systems and Applications, 1st ed. (Taylor & Francis, 1997).

http://sp.itme.edu.pl/PRdouble.htm.

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Figures (7)

Fig. 1
Fig. 1

Scheme of standard GS algorithm for determining the transmittance of a single DOE. u is the field, φ is the phase, and A is the amplitude. This algorithm is used when the target intensity distribution is at the far field. Then, the algorithm uses a fast Fourier transform (FFT) to propagate between the element plane and the output plane and the inverse fast Fourier transform (IFFT) to propagate between the out plane and the element plane.

Fig. 2
Fig. 2

Results for the conventional GS algorithm in the near-field approach, changing Fourier transform for FrT. (a) Target image. (b) Intensity distribution produced by the target image in free propagation. (c) Intensity distribution using an amplitude DOE and (d) Intensity distribution using a phase DOE. The incident field is a plane wave with wavelength λ = 650 nm . The distance of propagation is 6 mm , the number of pixels is 128, and the mask size is 500 μm . For this distance, the results does not approximate to the target intensity distribution.

Fig. 3
Fig. 3

Sketch of the double DOE system, with the parameters involved. The initial field impinges on DOE 1 (amplitude or phase). Then, the field propagates a distance z 0 and reaches DOE 2 (amplitude or phase). The resulting field is propagated a distance z 1 from the second DOE to the observation plane.

Fig. 4
Fig. 4

Algorithm developed to determine the modulation of the DOEs in the double DOE configuration. The dashed line indicates the incorporation of the modified DOE (algorithm 2), which decreases the MSE, producing a better replicated image. Also we can see the effects produced at each stage of the algorithm. The parameters are defined in the text.

Fig. 5
Fig. 5

(a) Mean square error (MSE) for the possibilities for one DOE (A and P) and for the double DOE configuration (AA, AP, PA, and PP) using the image given in Fig. 2a as target. The size of the target image is 4 mm , the number of pixels is 128, z 0 = 3 mm , z 1 = 3 mm . The amplitude-phase configuration produces the best results. (b) MSE, nondiffraction efficiency (NDE) and nonuniformity (NU) for the amplitude-phase configuration.

Fig. 6
Fig. 6

Examples of double DOE system with AP configuration. In the three examples, DOE 1 presents a high transmittance at locations where the irradiance at the target image is high or at neighbor locations.

Fig. 7
Fig. 7

Phase DOE ( DOE 2 ) produced with algorithm 1 (a) and algorithm 2 (b) for the case of 6b (UCM logo).

Tables (1)

Tables Icon

Table 1 Best Values of the Quality Parameters for Fig. 6. a

Equations (15)

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U 2 ( x , y ; z ) = A target ( x , y ) e j φ ( x , y ) random .
U 3 ( x , y ) = A 3 ( x , y ) e j φ 3 ( x , y ) = IFrT [ U 2 ( x , y ) ] ,
h ( x , y ; z ) = e j k z j λ z { exp [ j k 2 z ( x 2 + y 2 ) ] } .
DOE 2 ( x , y ) = e j φ 3 ( x , y ) ,
U 4 ( x , y , z 0 ) = A 4 ( x , y ) e j φ 4 ( x , y ) = IFrT [ U 3 ( x , y ) ] .
DOE 1 ( x , y ) = A 4 ( x , y ) .
U 0 ( x , y ) = DOE 1 ( x , y ) u incident ( x , y ) .
U 1 ( x , y , z 0 ) = A 1 ( x , y ) e j φ 1 ( x , y ) = FrT [ U 0 ( x , y ) ] ,
h ( x , y ; z ) = e j k z j λ z { exp [ j k 2 z ( x 2 + y 2 ) ] } .
U 1 ( x , y ; z ) = U 1 ( x , y ) DOE 2 ( x , y ) .
U 2 ( x , y ; z ) = A 2 ( x , y ) e j φ 2 ( x , y ) = FrT [ U 1 ( x , y ; z ) ] .
DOE ¯ 2 ( x , y ) = DOE 2 ( x , y ) e j φ 1 ( x , y ) , U 1 ( x , y ; z ) = U 1 ( x , y ) DOE ¯ 2 ( x , y ) .
MSE = | I target ( x , y ) k I output ( x , y ) | 2 d x d y | I target ( x , y ) | 2 d x d y ,
NDE = 1 s I output ( x , y ) d x d y I output ( x , y ) d x d y ,
NU = std [ I output ( x , y ) ] S ,

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