Abstract

An error diffusion algorithm is used to design binary pixelated beam shapers. Beam shapers based on pixels with transmission equal either to 0 or 1 can be synthesized with this deterministic algorithm to provide continuous intensity shaping after Fourier filtering. The capabilities of these shapers are studied for high-power lasers. A particular emphasis is placed on the influence of the feature size on the performance of the designed shapers, and it is shown that the transmission degradation is highly predictable for this algorithm. Simulations demonstrating the possibility of precompensating for feature size when designing intensity shapers are presented.

© 2007 Optical Society of America

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    [CrossRef]
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2006 (1)

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

2004 (1)

G. H. Miller, E. I. Moses, and C. R. Wuest, "The National Ignition Facility: enabling fusion ignition for the 21st century," Nucl. Fusion 44, S228-S238 (2004).
[CrossRef]

2002 (1)

2001 (2)

1997 (1)

1994 (1)

1992 (1)

1989 (1)

1988 (1)

R. A. Ulichney, "Dithering with blue noise," Proc. IEEE 76, 56-79 (1988).
[CrossRef]

1981 (1)

J. Stoffel and J. Moreland, "A survey of electronic techniques for pictorial image, reproduction," IEEE Trans. Commun. 29, 1898-1925 (1981).
[CrossRef]

1980 (1)

1976 (1)

R. W. Floyd and L. Steinberg, "An adaptive algorithm for spatial greyscale," J. Soc. Inf. Disp. 17, 75-77 (1976).

1974 (2)

G. Dubé, "Total internal reflection apodizers," Opt. Commun. 12, 344-347 (1974).
[CrossRef]

A. J. Campillo, B. Carpenter, B. E. Newnam, and S. L. Shapiro, "Soft apertures for reducing damage in high-power laser-amplifier systems," Opt. Commun. 10, 313-315 (1974).
[CrossRef]

Auerbac, J. M.

Bagnoud, V.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

V. Bagnoud, J. Luce, L. Videau, and A. Rouyer, "Diode-pumped regenerative amplifier delivering 100-mJ single-mode laser pulses," Opt. Lett. 26, 337-339 (2001).
[CrossRef]

Barker, C. E.

Begishev, I. A.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Behrendt, W. C.

Bromage, J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Browning, D. F.

Byer, R. L.

Campbell, J. H.

Campillo, A. J.

A. J. Campillo, B. Carpenter, B. E. Newnam, and S. L. Shapiro, "Soft apertures for reducing damage in high-power laser-amplifier systems," Opt. Commun. 10, 313-315 (1974).
[CrossRef]

Carpenter, B.

A. J. Campillo, B. Carpenter, B. E. Newnam, and S. L. Shapiro, "Soft apertures for reducing damage in high-power laser-amplifier systems," Opt. Commun. 10, 313-315 (1974).
[CrossRef]

Cherezova, T. Yu.

Chesnokov, S. S.

Dalton, S.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Dubé, G.

G. Dubé, "Total internal reflection apodizers," Opt. Commun. 12, 344-347 (1974).
[CrossRef]

Efron, U.

U. Efron, Spatial Light Modulator Technology: Materials, Devices, and Applications, Vol. 47 of Optical Engineering (Marcel Dekker, 1995).

Emiliani, G.

Floyd, R. W.

R. W. Floyd and L. Steinberg, "An adaptive algorithm for spatial greyscale," J. Soc. Inf. Disp. 17, 75-77 (1976).

Folnsbee, L.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Giuliani, G.

Guardalben, M. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Henesian, M.

M. Henesian, Lawrence Livermore National Laboratory, Livermore, Calif. 94550 (personal communication, 2006).

Jungquist, R.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Kaptsov, L. N.

Karpenko, V. P.

Kelly, J. H.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Kessler, T. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Kruschwitz, B. E.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Kudryashov, A. V.

Laporta, P.

Loucks, S. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Luce, J.

Madou, M. J.

M. J. Madou, Fundamentals of Microfabrication: The Science of Miniaturization (CRC Press, 2002).

Magni, V.

Matousek, P.

Maywar, D. N.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

McCrory, R. L.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Meyerhofer, D. D.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Miller, G. H.

G. H. Miller, E. I. Moses, and C. R. Wuest, "The National Ignition Facility: enabling fusion ignition for the 21st century," Nucl. Fusion 44, S228-S238 (2004).
[CrossRef]

Moreland, J.

J. Stoffel and J. Moreland, "A survey of electronic techniques for pictorial image, reproduction," IEEE Trans. Commun. 29, 1898-1925 (1981).
[CrossRef]

Morse, S. F. B.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Moses, E. I.

G. H. Miller, E. I. Moses, and C. R. Wuest, "The National Ignition Facility: enabling fusion ignition for the 21st century," Nucl. Fusion 44, S228-S238 (2004).
[CrossRef]

Murray, J. R.

New, G. H. C.

Newnam, B. E.

A. J. Campillo, B. Carpenter, B. E. Newnam, and S. L. Shapiro, "Soft apertures for reducing damage in high-power laser-amplifier systems," Opt. Commun. 10, 313-315 (1974).
[CrossRef]

Oliver, J. B.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Osvay, K.

Park, Y. K.

Piegrai, A.

Puth, J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Rigatti, A. L.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Ross, I. N.

Rouyer, A.

Samarkin, V. V.

Schmid, A. W.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Shapiro, S. L.

A. J. Campillo, B. Carpenter, B. E. Newnam, and S. L. Shapiro, "Soft apertures for reducing damage in high-power laser-amplifier systems," Opt. Commun. 10, 313-315 (1974).
[CrossRef]

Shoup, M. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Silvestri, S. De

Smith, I. C.

Speck, D. R.

Steinberg, L.

R. W. Floyd and L. Steinberg, "An adaptive algorithm for spatial greyscale," J. Soc. Inf. Disp. 17, 75-77 (1976).

Stoeckl, C.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Stoffel, J.

J. Stoffel and J. Moreland, "A survey of electronic techniques for pictorial image, reproduction," IEEE Trans. Commun. 29, 1898-1925 (1981).
[CrossRef]

Ulichney, R.

R. Ulichney, Digital Halftoning (MIT, 1987).

Ulichney, R. A.

R. A. Ulichney, "Dithering with blue noise," Proc. IEEE 76, 56-79 (1988).
[CrossRef]

Van Wonterghem, B. M.

Videau, L.

Waxer, L. J.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Weiner, D.

J. H. Kelly, L. J. Waxer, V. Bagnoud, I. A. Begishev, J. Bromage, B. E. Kruschwitz, T. J. Kessler, S. J. Loucks, D. N. Maywar, R. L. McCrory, D. D. Meyerhofer, S. F. B. Morse, J. B. Oliver, A. L. Rigatti, A. W. Schmid, C. Stoeckl, S. Dalton, L. Folnsbee, M. J. Guardalben, R. Jungquist, J. Puth, M. J. Shoup III, D. Weiner, and J. D. Zuegel, "OMEGA EP: high-energy petawatt capability for the OMEGA laser facility," J. Phys. IV 133, 75-80 (2006).

Weissbach, S.

Wuest, C. R.

G. H. Miller, E. I. Moses, and C. R. Wuest, "The National Ignition Facility: enabling fusion ignition for the 21st century," Nucl. Fusion 44, S228-S238 (2004).
[CrossRef]

Wyrowski, F.

Zuegel, J. D.

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Figures (10)

Fig. 1
Fig. 1

(a) Example of 40th-order super-Gaussian intensity; (b) example of intensity distribution required to precompensate for the spatial gain variation in OMEGA EP; (c) lineout of the intensity of (a) at the center of the beam along the x direction; (d) lineout of the intensity of (b) at the center of the beam along the x direction.

Fig. 2
Fig. 2

Principle of the generation of a continuous shaped intensity using a binary shaper and Fourier filtering.

Fig. 3
Fig. 3

Principle of the error diffusion algorithm, (a) Representation of the target shaper transmission t ( m , n ) ; (b) representation of the binary shaper transmission s ( m , n ) being designed; (c) chart describing the design process. The thick squares on (a) and (b) represent the pixels being processed. The horizontal arrows on (b) schematize the lexicographical process over the already processed pixels. The white arrows on (a) represent the error diffusion to adjacent nonprocessed pixels.

Fig. 4
Fig. 4

(a), (b), and (c) Close-ups of shapers designed with error diffusion for a target intensity transmission equal to 5%, 25%, and 75%, respectively. (d), (e), and (f) Close-ups of shapers designed with the random dither algorithm for a target intensity transmission equal to 5%, 25%, and 75%, respectively.

Fig. 5
Fig. 5

(a) Binary shaper corresponding to the transmission of Fig. 1b. (b) and (c) Close-ups of the binary pixel distribution at the center and at the upper right corner of the beam, respectively. (d) Lineout of the filtered intensity along the x direction (solid curve) and lineout of the difference between the filtered intensity and the target intensity (dashed curve).

Fig. 6
Fig. 6

Intensity of the far field of the binary beam shaper designed with the error diffusion algorithm (solid curve) and the random dither algorithm (squares) for the generation of a shaped intensity of Fig. 1b. The intensity is averaged over a 4 mrad angle in the y direction and is plotted versus the angle in the x direction.

Fig. 7
Fig. 7

(a) rms error for the realization of the shaped intensity of Fig. 1b as a function of the pinhole angular diameter in the case of the error diffusion algorithm (circles) and the random draw algorithm (squares). The solid curve corresponds to the propagation of the target electric field through the same filtering system, (b) rms error for the realization of the shaped intensity as a function of the pinhole angular diameter for a pixel size of 10 (solid curve), 20 (dashed curve), and 40 μ m (dotted curve) in the case of error diffusion.

Fig. 8
Fig. 8

Three lineouts of the simulated intensity of the shaped beam after amplification for shaping with (a) error diffusion and (b) random dither. In each case, the pixel size is 10 μ m , and the pinhole size leading to the smallest rms error is used.

Fig. 9
Fig. 9

Transmission error simulated as a function of the target intensity transmission for a size offset equal to 5% (lower curve) and 10% (upper curve). In each case, the circles correspond to the error prediction using Eq. (4), and the squares correspond to the error prediction using Eq. (6) (markers are plotted only in the domain of validity of these two equations).

Fig. 10
Fig. 10

Transmission error as a function of the target transmission for offsets equal to 4% (dotted curve), 5% (solid curve), and 6% (dashed curve) (a) without precompensation and (b) with precompensation of the pixel-size offset, assuming a pixel-size offset equal to 5%.

Equations (6)

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ε rms = y [ T obtained ( x , y ) T target ( x , y ) 1 ] 2 d x d y ,
ε peak = max s [ T obtained ( x , y ) T target ( x , y ) 1 ] ,
T obtained = d blank 2 = ( B S blank N S 0 ) 2 ,
T obtained = T target ( S blank S 0 ) 2 .
T obtained = ( 1 d metal ) 2 = ( 1 M S metal N S 0 ) 2 ,
T obtained = [ 1 ( 1 T target ) S metal S 0 ] 2 .

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