Abstract

We propose a method for designing arbitrary two-dimensional continuous phase elements. The phase anomalies that are due to amplitude zeros that occur in conventional design methods can be avoided completely with this method. Therefore the method is also useful for designing holographic elements that are free from amplitude zeros. The method is based on an iterative Fourier-transform algorithm. In the iteration process the method operates on the unwrapped phase. Consequently, a subsequent continuation step is not needed.

© 2000 Optical Society of America

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References

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  1. J. R. Leger, D. Chen, and Z. Wang, Opt. Lett. 19, 108 (1994).
    [CrossRef]
  2. J. N. Mait, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 293–324.
  3. U. Krackhardt, J. N. Mait, and N. Streibl, Appl. Opt. 31, 27 (1992).
    [CrossRef] [PubMed]
  4. F. Wyrowski, in Diffractive Optics, F. Wyrowski and J. Turunen, eds. (Akademie Verlag, Berlin, 1997).
  5. M. T. Gale, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 87–126.
  6. E. B. Kley, T. Posner, and R. Göring, Int. J. Optoelectron. 8, 513 (1993).
  7. R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
    [CrossRef]
  8. R.-P. Salmio, H. Saarikoski, and J. Saarinen, Opt. Lett. 22, 591 (1997).
    [CrossRef] [PubMed]
  9. J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.
  10. J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.
  11. J. Bähr and K.-H. Brenner, Appl. Opt. 35, 5102 (1996).
    [CrossRef]
  12. K.-H. Brenner and J. Bähr, in Technical Digest of the Sixth Microoptics Conference and the Fourteenth Topical Meeting on Gradient-Index Optical Systems (Japan Society of Applied Physics, Tokyo, 1997), p. 24.
  13. W. Singer and K.-H. Brenner, J. Opt. Soc. Am. A 12, 1913 (1995).
    [CrossRef]
  14. H. P. Herzig, D. Prongue, and R. Dändliker, Jpn. J. Appl. Phys. 29, 1307 (1990).
    [CrossRef]
  15. R. W. Gerchberg and W. O. Saxton, Optik 35, 237 (1972).
  16. J. R. Fienup, Opt. Eng. 19, 297 (1980).
    [CrossRef]
  17. F. Wyrowski and O. Bryngdahl, J. Opt. Soc. Am. A 5, 1058 (1988).
    [CrossRef]
  18. M. Totzeck and H. J. Tiziani, Opt. Commun. 138, 365 (1997).
    [CrossRef]

1997 (2)

1996 (1)

1995 (2)

W. Singer and K.-H. Brenner, J. Opt. Soc. Am. A 12, 1913 (1995).
[CrossRef]

R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
[CrossRef]

1994 (1)

1993 (1)

E. B. Kley, T. Posner, and R. Göring, Int. J. Optoelectron. 8, 513 (1993).

1992 (1)

1990 (1)

H. P. Herzig, D. Prongue, and R. Dändliker, Jpn. J. Appl. Phys. 29, 1307 (1990).
[CrossRef]

1988 (1)

1980 (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

1972 (1)

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

Bähr, J.

J. Bähr and K.-H. Brenner, Appl. Opt. 35, 5102 (1996).
[CrossRef]

K.-H. Brenner and J. Bähr, in Technical Digest of the Sixth Microoptics Conference and the Fourteenth Topical Meeting on Gradient-Index Optical Systems (Japan Society of Applied Physics, Tokyo, 1997), p. 24.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

Brenner, K.-H.

J. Bähr and K.-H. Brenner, Appl. Opt. 35, 5102 (1996).
[CrossRef]

W. Singer and K.-H. Brenner, J. Opt. Soc. Am. A 12, 1913 (1995).
[CrossRef]

K.-H. Brenner and J. Bähr, in Technical Digest of the Sixth Microoptics Conference and the Fourteenth Topical Meeting on Gradient-Index Optical Systems (Japan Society of Applied Physics, Tokyo, 1997), p. 24.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

Bryngdahl, O.

Chen, D.

Dändliker, R.

H. P. Herzig, D. Prongue, and R. Dändliker, Jpn. J. Appl. Phys. 29, 1307 (1990).
[CrossRef]

Fienup, J. R.

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Gale, M. T.

M. T. Gale, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 87–126.

Gerchberg, R. W.

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

Göring, R.

E. B. Kley, T. Posner, and R. Göring, Int. J. Optoelectron. 8, 513 (1993).

Herzig, H. P.

H. P. Herzig, D. Prongue, and R. Dändliker, Jpn. J. Appl. Phys. 29, 1307 (1990).
[CrossRef]

Kley, E. B.

E. B. Kley, T. Posner, and R. Göring, Int. J. Optoelectron. 8, 513 (1993).

Krackhardt, U.

Leger, J. R.

Mait, J. N.

U. Krackhardt, J. N. Mait, and N. Streibl, Appl. Opt. 31, 27 (1992).
[CrossRef] [PubMed]

J. N. Mait, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 293–324.

Moisel, J.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

Posner, T.

E. B. Kley, T. Posner, and R. Göring, Int. J. Optoelectron. 8, 513 (1993).

Prongue, D.

H. P. Herzig, D. Prongue, and R. Dändliker, Jpn. J. Appl. Phys. 29, 1307 (1990).
[CrossRef]

Saarikoski, H.

Saarinen, J.

R.-P. Salmio, H. Saarikoski, and J. Saarinen, Opt. Lett. 22, 591 (1997).
[CrossRef] [PubMed]

R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
[CrossRef]

Salmio, R.

R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
[CrossRef]

Salmio, R.-P.

Saxton, W. O.

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

Singer, W.

W. Singer and K.-H. Brenner, J. Opt. Soc. Am. A 12, 1913 (1995).
[CrossRef]

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

Sinzinger, S.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

Spick, T.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

Streibl, N.

Tervonen, A.

R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
[CrossRef]

Testorf, M.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

Tiziani, H. J.

M. Totzeck and H. J. Tiziani, Opt. Commun. 138, 365 (1997).
[CrossRef]

Totzeck, M.

M. Totzeck and H. J. Tiziani, Opt. Commun. 138, 365 (1997).
[CrossRef]

Turunen, J.

R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
[CrossRef]

Wang, Z.

Wyrowski, F.

F. Wyrowski and O. Bryngdahl, J. Opt. Soc. Am. A 5, 1058 (1988).
[CrossRef]

F. Wyrowski, in Diffractive Optics, F. Wyrowski and J. Turunen, eds. (Akademie Verlag, Berlin, 1997).

Appl. Opt. (2)

Appl. Phys. Lett. (1)

R. Salmio, J. Saarinen, J. Turunen, and A. Tervonen, Appl. Phys. Lett. 66, 917 (1995).
[CrossRef]

Int. J. Optoelectron. (1)

E. B. Kley, T. Posner, and R. Göring, Int. J. Optoelectron. 8, 513 (1993).

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

Jpn. J. Appl. Phys. (1)

H. P. Herzig, D. Prongue, and R. Dändliker, Jpn. J. Appl. Phys. 29, 1307 (1990).
[CrossRef]

Opt. Commun. (1)

M. Totzeck and H. J. Tiziani, Opt. Commun. 138, 365 (1997).
[CrossRef]

Opt. Eng. (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).
[CrossRef]

Opt. Lett. (2)

Optik (1)

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

Other (6)

K.-H. Brenner and J. Bähr, in Technical Digest of the Sixth Microoptics Conference and the Fourteenth Topical Meeting on Gradient-Index Optical Systems (Japan Society of Applied Physics, Tokyo, 1997), p. 24.

J. N. Mait, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 293–324.

F. Wyrowski, in Diffractive Optics, F. Wyrowski and J. Turunen, eds. (Akademie Verlag, Berlin, 1997).

M. T. Gale, in Micro-Optics, Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 87–126.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Proceedings of the International Commission for Optics Topical Meeting on Optical Computing (Russian Academy of Sciences, Minsk, 1992), p. 234.

J. Bähr, K.-H. Brenner, J. Moisel, W. Singer, S. Sinzinger, T. Spick, and M. Testorf, in Technical Digest of the Tenth Topical Meeting on Gradient-Index Optical Systems (European Optical Society, Orsay, France, 1992), p. 187.

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Figures (3)

Fig. 1
Fig. 1

(a) Wrapped phase for the 9×9 array generator element, (b) unwrapped phase ranging from 0 (black) to 3.6π (white).

Fig. 2
Fig. 2

Array generator reconstructed from the phase shown in Fig. 1.

Fig. 3
Fig. 3

(a) Unwrapped phase for the letter F, ranging from 0 (black) to 4.05π (white). (b) Amplitude reconstructed from the phase in (a).

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

ϕx,y=wrapφx,y,2π,
wrapX,A=X-A1+floorXA-12-A/2A/2.
φx,y=T-1φ~v,μHv,μ, Hv,μ=R/ρifρ>R1otherwise,
In+1x,y=αInx,y+1-αIdx,y,
An+1x,yexpiϕn+1x,y=Ren+1x,y+iImn+1x,y.
φn+1x,y=φnx,y+wrapϕn+1x,y-φnx,y,2π.

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