Abstract

A new, to our knowledge, iterative algorithm for achieving optimization of beam profiles in a three-dimensional volume is presented. The algorithm is based on examining the region of interest at discrete plane locations perpendicular to the propagation direction. At each such plane an intensity constraint is imposed within a well-defined transverse spatial region of interest, whereas the phase inside that region as well as the complex amplitude outside the region is left unchanged from the previous iteration. Once the optimal solution is found, the mask that generates the desired distribution can be readily implemented with a planar diffractive optical element such as a computer-generated hologram. Several computer simulations verified the utility of the proposed approach.

© 1999 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions of nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  2. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989).
    [CrossRef] [PubMed]
  3. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  4. J. Rosen, “Synthesis of nondiffracting beams in free space,” Opt. Lett. 19, 369–371 (1994).
    [PubMed]
  5. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]
  6. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1749 (1967).
    [CrossRef] [PubMed]
  7. D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding techniques for the design of zero-order (on–axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
    [CrossRef]
  8. D. Mendlovic, Z. Zalevsky, G. Shabtay, E. Marom, “High-efficiency arbitrary array generator,” Appl. Opt. 35, 6875–6880 (1996).
    [CrossRef] [PubMed]
  9. E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
    [CrossRef]
  10. T. Haist, M. Schönleber, H. J. Tiziani, “Computer-generated holograms from 3D objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 66.
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determining of phase from image and diffraction plane figures,” Optik 35, 237–246 (1972).

1997

D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding techniques for the design of zero-order (on–axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
[CrossRef]

T. Haist, M. Schönleber, H. J. Tiziani, “Computer-generated holograms from 3D objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

1996

1994

1989

1987

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

J. Durnin, “Exact solutions of nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

1972

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

1967

Durnin, J.

Friberg, A. T.

Gerchberg, R. W.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 66.

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Haist, T.

T. Haist, M. Schönleber, H. J. Tiziani, “Computer-generated holograms from 3D objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Katz, J.

E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
[CrossRef]

Konforti, N.

E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
[CrossRef]

Levy, U.

Lohmann, A. W.

Marom, E.

D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding techniques for the design of zero-order (on–axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, G. Shabtay, E. Marom, “High-efficiency arbitrary array generator,” Appl. Opt. 35, 6875–6880 (1996).
[CrossRef] [PubMed]

E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
[CrossRef]

Mendlovic, D.

D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding techniques for the design of zero-order (on–axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, G. Shabtay, E. Marom, “High-efficiency arbitrary array generator,” Appl. Opt. 35, 6875–6880 (1996).
[CrossRef] [PubMed]

E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Paris, D. P.

Piestun, R.

Rosen, J.

Saxton, W. O.

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

Schönleber, M.

T. Haist, M. Schönleber, H. J. Tiziani, “Computer-generated holograms from 3D objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Shabtay, G.

Shamir, J.

Spektor, B.

Tan, C.

E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
[CrossRef]

Tiziani, H. J.

T. Haist, M. Schönleber, H. J. Tiziani, “Computer-generated holograms from 3D objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Turunen, J.

Vasara, A.

Zalevsky, Z.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

T. Haist, M. Schönleber, H. J. Tiziani, “Computer-generated holograms from 3D objects written on twisted-nematic liquid crystal displays,” Opt. Commun. 140, 299–308 (1997).
[CrossRef]

Opt. Lett.

Optik

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

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1996), p. 66.

E. Marom, D. Mendlovic, N. Konforti, J. Katz, C. Tan, “Diffractive optic elements for forming scanning beams,” in Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich, S. H. Lee, eds., Proc. SPIE3010, 30–37 (1997).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic of the optical setup including the volume of interest the (VOI).

Fig. 2
Fig. 2

Block diagram of the proposed algorithm.

Fig. 3
Fig. 3

Profiles at various longitudinal cross sections: crosses, the profile obtained by the algorithm; open circles, the Gaussian profile. (a) Z = Z 0, (b) Z = Z 0 + 0.45 m, (c) Z = Z 0 + 0.9 m, (d) Z = Z 0 + 1.35 m.

Fig. 4
Fig. 4

Image of the profile obtained by the algorithm versus propagation distance.

Fig. 5
Fig. 5

Same as Fig. 4 for a conventional Gaussian beam with the same waist.

Fig. 6
Fig. 6

MTF obtained at eight equally spaced longitudinal planes (Z 0 - 0.4: Z 0 + 0.4 m).

Equations (6)

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uix, y, zi=zijλ  u¯x, y, 0×expjkzi2+x-x2+y-y21/2zi2+x-x2+y-y2×dxdy,  i1, 2,, N,
u¯x, y, zi=ηgx, y, ziexp(jphaseux, y, zi)x, yROux, y, ziotherwise,
uix, y, 0=-zijλ  u¯x, y, zi×exp-jkzi2+x-x2+y-y21/2zi2+x-x2+y-y2×dxdy.
ulx, y, 0=i=1N wiuix, y, 0,
MTFux, y=FT- |ux, y|2dyFT- |u0, y|2dy,
|ur|=exp-rw2J02πλ ar,

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