Abstract

Algorithms based on the fast Fourier transform (FFT) for the design of spot-generating computer generated holograms (CGHs) typically only make use of a few sample positions in the propagated field. We have developed a new design method that much better utilizes the information-carrying capacity of the sampled propagated field. In this way design tasks which are difficult to accomplish with conventional FFT-based design methods, such as spot positioning at non-sample positions and/or spot positioning in 3D, are solved as easily as any standard design task using a conventional method. The new design method is based on a projection optimization, similar to that in the commonly used Gerchberg-Saxton algorithm, and the vastly improved design freedom comes at virtually no extra computational cost compared to the conventional design. Several different design tasks were demonstrated experimentally with a liquid crystal spatial light modulator, showing highly accurate creation of the desired field distributions.

© 2009 Optical Society of America

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    [CrossRef]
  27. T. Haist, M. Schoenleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
    [CrossRef]
  28. M. ˇSkeren, I. Richter, and P. Fiala, "Design and optimization considerations of multi-focus phase-only diffractive elements," Proc. SPIE 5182, 236-245 (2004).
  29. J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
    [CrossRef]
  30. R. D. Leonardo, F. Ianni, and G. Ruocco, "Computer generation of optimal holograms for optical trap arrays," Opt. Express 15, 1913-1922 (2006).
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    [CrossRef] [PubMed]
  32. H. Akahori, "Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform," Appl. Opt. 25, 802-811 (1986).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

2009

2008

2007

2006

2004

M. ˇSkeren, I. Richter, and P. Fiala, "Design and optimization considerations of multi-focus phase-only diffractive elements," Proc. SPIE 5182, 236-245 (2004).

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

J. Enderlein and F. Pampaloni, "Unified operator approach for deriving Hermite-Gaussian and Laguerre-Gaussian laser modes," J. Opt. Soc. Am. A 21, 1553-1558 (2004).
[CrossRef]

2003

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

2002

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

M. Skeren, I. Richter, and P. Fiala, "Iterative Fourier transform algorithm: comparison of various approaches," J. Mod. Opt. 49, 1851-1870 (2002).
[CrossRef]

J. S. Liu and M. R. Taghizadeh, "Iterative algorithm for the design of diffractive phase elements for laser beam shaping," Opt. Lett. 27, 1463-1465 (2002).
[CrossRef]

2001

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

1999

1997

T. Haist, M. Schoenleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

1996

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Iterative calculation of diffractive optical elements focusing into three-dimensional domain and onto the surface of the body of rotation," J. Mod. Opt. 43, 1509-1524 (1996).
[CrossRef]

J. Bengtsson, N. Eriksson, and A. Larsson, "Small-feature-size fan-out kinoform etched in GaAs," Appl. Opt. 35, 801-806 (1996).
[CrossRef] [PubMed]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

1994

1991

1990

1989

1988

1987

1986

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).

Akahori, H.

Allebach, J. P.

Bengtsson, J.

Bryngdahl, O.

Byckling, E.

Corkum, D. L.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Dames, M. P.

Dearing, M. T.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Dorschner, T. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Dowling, R. J.

Dufresne, E. R.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Enderlein, J.

Engstr¨om, D.

Engstrom, D.

Eriksson, E.

Eriksson, N.

Fagerholm, J.

Farn, M. W.

M. W. Farn, "New iterative algorithm for the design of phase-only gratings," Proc. SPIE 1555, 34-42 (1991).
[CrossRef]

Feldman, M. R.

Friedman, L. J.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Galt, S.

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).

Goksor, M.

Grier, D. G.

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Guest, C. C.

Haist, T.

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
[CrossRef]

T. Haist, M. Schoenleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Handschy, M. A.

Heikonen, J.

Herzig, H. P.

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Hobbs, D. S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Holtz, M.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Ianni, F.

Jennison, B. K.

Kajanto, M.

Kettunen, V.

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Iterative calculation of diffractive optical elements focusing into three-dimensional domain and onto the surface of the body of rotation," J. Mod. Opt. 43, 1509-1524 (1996).
[CrossRef]

Konforti, N.

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Iterative calculation of diffractive optical elements focusing into three-dimensional domain and onto the surface of the body of rotation," J. Mod. Opt. 43, 1509-1524 (1996).
[CrossRef]

Larsson, A.

Leonardo, R. D.

Liberman, S.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Liu, J. S.

Marom, E.

McKee, P.

McManamon, P. F.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Milewski, G.

Nguyen, H. Q.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

O’Callaghan, M. J.

Pampaloni, F.

Reicherter, M.

Resler, D. P.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Ripoll, O.

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[CrossRef]

Ruocco, G.

Salin, A.

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).

Schoenleber, M.

T. Haist, M. Schoenleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Scrimgeour, J.

Seldowitz, M. A.

Sharp, R. C.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Sheets, S. A.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Iterative calculation of diffractive optical elements focusing into three-dimensional domain and onto the surface of the body of rotation," J. Mod. Opt. 43, 1509-1524 (1996).
[CrossRef]

Spalding, G. C.

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, "Computer-generated holographic optical tweezer arrays," Rev. Sci. Instrum. 72, 1810-1816 (2001).
[CrossRef]

Sweeney, D. W.

Taghizadeh, M. R.

Tiziani, H. J.

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, "Optical particle trapping with computer-generated holograms written on a liquid-crystal display," Opt. Lett. 24, 608-610 (1999).
[CrossRef]

T. Haist, M. Schoenleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Turunen, J.

Vasara, A.

Wagemann, E. U.

Walker, C.

Watson, E. A.

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

Wood, D.

Wyrowski, F.

Yatagai, T.

Yoshikawa, N.

Appl. Opt.

M. Kajanto, E. Byckling, J. Fagerholm, J. Heikonen, J. Turunen, A. Vasara, and A. Salin, "Photolithographic fabrication method of computer-generated holographic interferograms," Appl. Opt. 28, 778-784 (1989).
[CrossRef] [PubMed]

J. Bengtsson, N. Eriksson, and A. Larsson, "Small-feature-size fan-out kinoform etched in GaAs," Appl. Opt. 35, 801-806 (1996).
[CrossRef] [PubMed]

D. Engstrom, M. J. O’Callaghan, C. Walker, and M. A. Handschy, "Fast beam steering with a ferroelectric-liquidcrystal optical phased array," Appl. Opt. 48, 1721-1726 (2009).
[CrossRef] [PubMed]

M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, "Synthesis of digital holograms by direct binary search," Appl. Opt. 26, 2788-2798 (1987).
[CrossRef] [PubMed]

G. Milewski, D. Engstrm, and J. Bengtsson, "Diffractive optical elements designed for highly precise far-field generation in the presence of artifacts typical for pixelated spatial light modulators," Appl. Opt. 46, 95-105 (2007).
[CrossRef]

M. P. Dames, R. J. Dowling, P. McKee, and D. Wood, "Efficient optical elements to generate intensity weighted spot arrays: design and fabrication," Appl. Opt. 30, 2685-2691 (1991).
[CrossRef] [PubMed]

N. Yoshikawa, and T. Yatagai, "Phase optimization of a kinoform by simulated annealing," Appl. Opt. 33, 863-868 (1994).
[CrossRef] [PubMed]

J. Bengtsson, "Kinoform design with an optimal-rotation-angle method," Appl. Opt. 33, 6879-6884 (1994).
[CrossRef] [PubMed]

H. Akahori, "Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform," Appl. Opt. 25, 802-811 (1986).
[CrossRef] [PubMed]

D. Engstrom, G. Milewski, J. Bengtsson, and S. Galt, "Diffraction-based determination of the phase modulation for general spatial light modulators," Appl. Opt. 45, 7195-7204 (2006).
[CrossRef] [PubMed]

J. Mod. Opt.

M. Skeren, I. Richter, and P. Fiala, "Iterative Fourier transform algorithm: comparison of various approaches," J. Mod. Opt. 49, 1851-1870 (2002).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Iterative calculation of diffractive optical elements focusing into three-dimensional domain and onto the surface of the body of rotation," J. Mod. Opt. 43, 1509-1524 (1996).
[CrossRef]

J. Opt. Soc. Am. A

Nature

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Opt. Commun.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

T. Haist, M. Schoenleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Opt. Eng.

O. Ripoll, V. Kettunen, and H. P. Herzig, "Review of iterative Fourier-transform algorithms for beam shaping applications," Opt. Eng. 43, 2549-2556 (2004).
[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).

Proc. SPIE

M. W. Farn, "New iterative algorithm for the design of phase-only gratings," Proc. SPIE 1555, 34-42 (1991).
[CrossRef]

P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs, M. Holtz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson, "Optical phased array technology," Proc. SPIE 84, 268-298 (1996).

M. ˇSkeren, I. Richter, and P. Fiala, "Design and optimization considerations of multi-focus phase-only diffractive elements," Proc. SPIE 5182, 236-245 (2004).

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[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

(a) An SLM and its output field consisting of two spots positioned at different distances from the SLM. The iteration plane (IP) and the optimization regions are indicated. Typical (b) amplitude and (c) phase of Edesired n for an optimization region.

Fig. 2.
Fig. 2.

(a) Flowchart of the algorithm. (b) Phase read-out and reshaping procedure for a single optimization region. (c) Amplitude modification scheme for the IP-field. Note that the amplitude is shown in false scale to enhance the visibility of the unwanted ghost spots.

Fig. 3.
Fig. 3.

Schematic free-space setup. Lenses L1 and L2 expand the HeNe laser beam. Polarizer P is oriented such that the SLM yields phase-only modulation. The intensity distribution at some distance from the SLM is captured with a CCD camera. Inset shows the 256×256 pixels of a calculated CGH (phase pattern) displayed on the SLM.

Fig. 4.
Fig. 4.

(a) The desired 5×5 spot pattern with movement directions of the four moving spots indicated. (b) The evolution of the efficiency η and the uniformity error ε of the power in the IP and SP for the design of one of the 25 CGHs. Note that the uniformity error in the IP and SP are close to identical. Simulated (c) and measured (d) relative positions for the 25 spots of the 25 CGHs.

Fig. 5.
Fig. 5.

Desired (a) amplitude and (b) phase in the IP (positioned 46 cm from the SLM) for Laguerre-Gaussian of (0,1) order (outer circle) and Gaussian (inner circle) shaped spots, focused in a plane 50 cm from the SLM. (c) The evolution of the efficiency η and the uniformity error ε calculated from the power distribution in the IP and SP. (d) Spatial phase modulation of the optimized CGH. (e) Simulated and (f) measured intensity distribution in the spot plane.

Fig. 6.
Fig. 6.

(a) Schematic 3D design task (not to scale); 25 spots positioned in groups of five at z 1=45.0 to z 5=55.0 cm using z IP=49.0 cm. (b) Amplitude and (c) phase of the desired field in the IP. (d) The evolution of the efficiency η and the uniformity error ε based on the power distribution in the IP and the power in the actual spot positions. (e) Measured intensity distributions in the planes, parallel to the SLM, containing the spots. Note that the intensity is shown in a logarithmic scale to enhance the visibility of the defocused spots.

Equations (1)

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wn=wn1(PndesiredPnobtained)c.

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