Abstract

One of the most important factors that limit the performance of diffractive optical elements (DOE’s) is the depth accuracy of the relief structure. A common procedure for fabricating DOE’s is the binary optics procedure, in which binary masks are used for the fabrication of a multilevel relief structure. Here an analytic procedure for calculating the optimal depth levels of DOE’s, the phase bias, and the decision levels is presented. This approach is based on the minimization of the mean-squared error caused by the quantization of the continuous profile. As a result of the minimization an optimal value for the etching depth of each photolithographic mask is determined. The obtained depth values are, in general, different from the depth values used by the conventional multilevel approach. Comprehensive mathematical analysis is given, followed by several computer simulations that demonstrate the advantages of the proposed procedure.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am A 7, 961–969 (1988).
    [CrossRef]
  2. J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filters,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  3. S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
    [CrossRef]
  4. M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
    [CrossRef]
  5. H. M. Phillips, R. A. Sauerbrey, “Eximer-laser-produced nanostructures in polymers,” Opt. Eng. 32, 2424–2436 (1993).
    [CrossRef]
  6. N. A. Vainos, S. Mailis, S. Pissadakis, L. Boutsikaris, P. J. M. Parmiter, P. Dainty, T. J. Hall, “Excimer laser use for microetching computer-generated holographic structures,” Appl. Opt. 35, 6304–6319 (1996).
    [CrossRef] [PubMed]
  7. M. Ekberg, M. Larsson, S. Hard, B. Nilsson, “Multilevel phase holograms manufactured by electron-beam lithography,” Opt. Lett. 15, 568–569 (1990).
    [CrossRef] [PubMed]
  8. M. Larsson, M. Ekberg, F. Nikolajeff, S. Hard, “Successive-development optimization of resist kinoforms manufactured with direct writing, electron-beam lithography,” Appl. Opt. 33, 1176–1179 (1994).
    [CrossRef] [PubMed]
  9. H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
  10. G. J. Swanson, W. B. Weldkamp, “High-efficiency, multilevel, diffractive optical elements,” US. patent4,895,790 (23January1987).
  11. W. H. Welch, J. E. Morris, M. R. Feldman, “Iterative discrete on-axis encoding of radially symmetric computer-generated holograms,” J. Opt. Soc. Am. A 10, 1729–1738 (1993).
    [CrossRef]
  12. M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
    [CrossRef] [PubMed]
  13. J. Fan, D. Zaleta, K. S. Urquhart, S. H. Lee, “Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication,” Appl. Opt. 34, 2522–2533 (1995).
    [CrossRef] [PubMed]
  14. C. Chen, A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
    [CrossRef]
  15. V. Arrizon, S. Sinzinger, “Modified quantization schems for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
    [CrossRef]
  16. A. K. Jain, Fundamentals of Digital Image Processing (Prentice-HallEngelwood Cliffs, N.J., 1989).
  17. J. Max, “Quantizing for minimum distortion,” IEEE Trans. Inf. Theory IT-6, 7–12 (1960).
    [CrossRef]
  18. N. C. Gallagher, “Optimum quantization in digital holography,” Appl. Opt. 17, 109–115 (1978).
    [CrossRef] [PubMed]
  19. G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
    [CrossRef]
  20. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1997 (2)

V. Arrizon, S. Sinzinger, “Modified quantization schems for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

C. Chen, A. A. Sawchuk, “Nonlinear least-squares and phase-shifting quantization methods for diffractive optical element design,” Appl. Opt. 36, 7297–7306 (1997).
[CrossRef]

1996 (1)

1995 (2)

1994 (2)

M. Larsson, M. Ekberg, F. Nikolajeff, S. Hard, “Successive-development optimization of resist kinoforms manufactured with direct writing, electron-beam lithography,” Appl. Opt. 33, 1176–1179 (1994).
[CrossRef] [PubMed]

M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

1993 (2)

1992 (1)

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

1990 (1)

1989 (1)

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

1988 (1)

F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am A 7, 961–969 (1988).
[CrossRef]

1985 (1)

1978 (1)

1972 (1)

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

1960 (1)

J. Max, “Quantizing for minimum distortion,” IEEE Trans. Inf. Theory IT-6, 7–12 (1960).
[CrossRef]

Arrizon, V.

V. Arrizon, S. Sinzinger, “Modified quantization schems for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Boutsikaris, L.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am A 7, 961–969 (1988).
[CrossRef]

Chen, C.

Dainty, P.

Ekberg, M.

Fan, J.

Feldman, M. R.

Gale, M. T.

M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Gallagher, N. C.

Gerchberg, R. W.

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

Hall, T. J.

Hard, S.

Herzig, H. P.

M. Kuittinen, H. P. Herzig, “Encoding of efficient diffractive microlenses,” Opt. Lett. 20, 2156–2158 (1995).
[CrossRef] [PubMed]

H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

Horner, J. L.

Jahns, J.

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-HallEngelwood Cliffs, N.J., 1989).

Kuittinen, M.

Larsson, M.

Lee, S. H.

Leger, J. R.

Mailis, S.

Max, J.

J. Max, “Quantizing for minimum distortion,” IEEE Trans. Inf. Theory IT-6, 7–12 (1960).
[CrossRef]

Morris, J. E.

Nikolajeff, F.

Nilsson, B.

Parmiter, P. J. M.

Pederson, J.

M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Phillips, H. M.

H. M. Phillips, R. A. Sauerbrey, “Eximer-laser-produced nanostructures in polymers,” Opt. Eng. 32, 2424–2436 (1993).
[CrossRef]

Pissadakis, S.

Rossi, M.

M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Sauerbrey, R. A.

H. M. Phillips, R. A. Sauerbrey, “Eximer-laser-produced nanostructures in polymers,” Opt. Eng. 32, 2424–2436 (1993).
[CrossRef]

Sawchuk, A. A.

Saxton, W. O.

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

Schutz, H.

M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Sinzinger, S.

V. Arrizon, S. Sinzinger, “Modified quantization schems for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Swanson, G. J.

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

G. J. Swanson, W. B. Weldkamp, “High-efficiency, multilevel, diffractive optical elements,” US. patent4,895,790 (23January1987).

Urquhart, K. S.

Vainos, N. A.

Walker, S. J.

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

Welch, W. H.

Weldkamp, W. B.

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

G. J. Swanson, W. B. Weldkamp, “High-efficiency, multilevel, diffractive optical elements,” US. patent4,895,790 (23January1987).

Wyrowski, F.

F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am A 7, 961–969 (1988).
[CrossRef]

Zaleta, D.

Appl. Opt. (6)

IEEE Trans. Inf. Theory (1)

J. Max, “Quantizing for minimum distortion,” IEEE Trans. Inf. Theory IT-6, 7–12 (1960).
[CrossRef]

J. Opt. Soc. Am A (1)

F. Wyrowski, O. Bryngdahl, “Iterative Fourier transform algorithm applied to computer holography,” J. Opt. Soc. Am A 7, 961–969 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

V. Arrizon, S. Sinzinger, “Modified quantization schems for Fourier-type array generator,” Opt. Commun. 140, 309–315 (1997).
[CrossRef]

Opt. Eng. (3)

G. J. Swanson, W. B. Weldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. 28, 605–608 (1989).
[CrossRef]

M. T. Gale, M. Rossi, J. Pederson, H. Schutz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

H. M. Phillips, R. A. Sauerbrey, “Eximer-laser-produced nanostructures in polymers,” Opt. Eng. 32, 2424–2436 (1993).
[CrossRef]

Opt. Lett. (2)

Optik (1)

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

Other (3)

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-HallEngelwood Cliffs, N.J., 1989).

H. P. Herzig, Micro-optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

G. J. Swanson, W. B. Weldkamp, “High-efficiency, multilevel, diffractive optical elements,” US. patent4,895,790 (23January1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Continuous (solid curve), uniform-binary-quantized (dashed line), and nonuniform-binary-quantized (curve with asterisks) lens profiles.

Fig. 2
Fig. 2

Image cross section at the focal plane, obtained with the uniform- (dashed curve) and the nonuniform- (solid curve) binary-quantized lenses.

Fig. 3
Fig. 3

Continuous (solid curve), uniform-binary-quantized (dashed line), and nonuniform-binary-quantized (curve with asterisks) lens (F = 5 m) profiles.

Fig. 4
Fig. 4

Probability-density function for the L DOE.

Fig. 5
Fig. 5

Continuous (solid curve), uniform-binary-quantized (dashed line), and nonuniform-binary-quantized (curve with asterisks) profiles of the L DOE.

Tables (6)

Tables Icon

Table 1 Optimal Etching Depth Levels for the Fabrication of the F = 8 m Fresnel Lens

Tables Icon

Table 2 Obtained MSE and Efficiency Values for the F = 8 m Fresnel Lens

Tables Icon

Table 3 Optimal Etching Depth Levels for the Fabrication of the F = 5 m Fresnel Lens

Tables Icon

Table 4 Obtained MSE and Efficiency Values for the F = 5 m Fresnel Lens

Tables Icon

Table 5 Optimal Etching Depth Levels for the Fabrication of the L DOE

Tables Icon

Table 6 Obtained MSE Values for the L DOE

Equations (14)

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

E=l=1l=2NClCl+1 |expiϕ-expiϕl|2pϕdϕ,
ECm=|expiCm-expiϕm-1|2pϕm-|expiCm-expiϕm|2pϕm=0,
Edr=n=12NEϕnϕndr=n=12NCnCn+1 sinϕ-ϕn×pϕϕdrdϕ=0,
Eϕ1=C1C2 sinϕ-ϕ1pϕdϕ=0,
Cm=ϕm+ϕm-1/2,
ϕn=ϕ1+r=1N drp=12r-1j=2N-r+12N-r+1 δn,j+p-12N-r+1,
δi,j=1 if i=j0 if i  j.
dr>j=r+1N dj, , 1<r<N-1.
ϕn=ϕ1+n-1N,2·d1, d2, , dNt,
ϕ1=ϕ1,  ϕ2=ϕ1+d2,  ϕ3=ϕ1+d1,  ϕ4=ϕ1+d1+d2;
ϕndr=00011011,
pϕ=constdxdϕ.
sinϕ-ϕnϕ-ϕn
ηN=sinc12N2.

Metrics