Abstract

The synthesis of three-dimensional (3-D) light distributions is important for many applications. For example, in scanning applications it is preferable that the scanning beam preserve its characteristics over a large distance to yield elongated scanning range. It is evident that any 3-D light distribution must satisfy the wave equation or, in second-order approximation, the Fresnel diffraction formula. Thus many desirable 3-D light distributions may not be realizable. We propose a single optical element (OE) that synthesizes a physical beam within a certain 3-D region. The OE provides the optimal physical beam in comparison with a desired one in the sense of minimal mean-square error.

© 2000 Optical Society of America

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References

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  1. J. R. Leger, in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), Chap. 9, pp. 223–257.
  2. Z. Zalevsky, D. Mendlovic, and R. Dorsch, Opt. Lett. 21, 842 (1996).
    [CrossRef] [PubMed]
  3. D. Mendlovic, Z. Zalevsky, G. Shabtay, and E. Marom, Appl. Opt. 35, 6875 (1996).
    [CrossRef] [PubMed]
  4. R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).
  5. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  6. Z. Jiang, J. Opt. Soc. Am. A 14, 1478 (1997).
    [CrossRef]
  7. J. Rosen, Opt. Lett. 19, 369 (1995).

1997 (1)

1996 (2)

1995 (1)

1987 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).

Other (1)

J. R. Leger, in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), Chap. 9, pp. 223–257.

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Figures (6)

Fig. 1
Fig. 1

The optical setup.

Fig. 2
Fig. 2

Cross sections of the amplitude profile of the desired beam for z=1,2,3,4 m (top left, top right, bottom left, bottom right, respectively).

Fig. 3
Fig. 3

Same as Fig. 2 but for a conventional Gaussian beam.

Fig. 4
Fig. 4

Same as Fig. 2 but for the beam obtained with the suggested filter.

Fig. 5
Fig. 5

Cross sections of the amplitude profile as a function of the distance: (a) the desired beam, (b) a conventional Gaussian beam, (c) the result obtained with the suggested filter.

Fig. 6
Fig. 6

Cross sections of the amplitude profile as a function of distance: (a) a beam with a rectangular aperture in middle range, (b) the result obtained with the suggested filter.

Equations (13)

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=--Dzfx,yhx,y;z-udx,y,z2dzdxdy,
=--Dzf˜u,vh˜u,v;z-u˜du,v,z2dzdudv,
f˜RDz|(f˜Ru,v+if˜1u,vh˜u,v;z-u˜du,v,z|2dz=0,
f˜1Dz|(f˜Ru,v+if˜1u,vh˜u,v;z-u˜du,v,z|2dz=0.
Dzh˜u,v;zf˜u,vh˜u,v;z-u˜du,v,z*dz+Dzh˜*u,v;zf˜u,vh˜u,v;z-u˜du,v,zdz=0,
iDzh˜u,v;zf˜u,vh˜u,v;z-u˜du,v,z*dz--iDzh˜*u,v;zf˜u,vh˜u,v;z-u˜du,v,zdz=0.
f˜u,v=Dzu˜du,v,zh˜*u,v;zdzDzh˜u,v;z2dz.
h˜u,v;z=expi2πλzexp-iπλzu2+v2,
f˜u,v=1ΔzDzu˜du,v,zexp-i2πλz×expiπλzu2+v2dz,
u˜du,v,z=exp-πa2u2+v2expiπλu2+v2z24×expi2πλzexp-iπλu2+v2z2.
f˜u,v=4exp-πa2u2+v2πλu2+v2z2expiπλz22u2+v2×sinπλz24u2+v2.
u˜du,v,z=Δ2 expi2πλzsincΔusincΔv.
f˜u,v=2Δ2 sincΔusincΔvπλz2u2+v2expiπλz22u2+v2×sinπλz22u2+v2.

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