Abstract

The intensity uniformity of the spots generated by fan-out diffractive optical elements (DOEs) (or kinoforms) is often highly sensitive to any fabrication error that leads to a deviation of the surface-relief depth of the DOE from its design value. Many of the fabrication errors, such as those that are due to insufficient control of development or etch rates, increase almost linearly with the desired relief depth in every position of the DOE. We present an algorithm for designing fan-out DOEs with a significantly reduced sensitivity of the intensity uniformity to such errors. The reduced sensitivity can be obtained without reducing the efficiency of the DOE. Experimental results for fabricated DOEs show that reduced sensitivity is also obtained in practice.

© 2002 Optical Society of America

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References

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  1. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  2. M. Duparre, M. A. Golub, B. Ludge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2-laser beams,” Appl. Opt. 34, 2489–2497 (1995).
    [CrossRef]
  3. H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
    [CrossRef]
  4. T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption,” Appl. Opt. 35, 6865–6874 (1996).
    [CrossRef] [PubMed]
  5. M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
    [CrossRef]
  6. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  7. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  8. M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongue, “Continuous-relief diffractive optical elements for two-dimensional array generation,” Appl. Opt. 32, 2526–2533 (1993).
    [CrossRef] [PubMed]
  9. G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
    [CrossRef] [PubMed]
  10. J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. 36, 8435–8444 (1997).
    [CrossRef]
  11. 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]
  12. P. Ehbets, M. Rossi, H.-P. Herzig, “Continuous-relief fan-out elements with optimized fabrication tolerances,” Opt. Eng. 34, 3456–3464 (1995).
    [CrossRef]
  13. G. Z. Yang, B. Y. Gu, X. Tan, M. P. Chang, B. H. Dong, O. K. Ersoy, “Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions,” J. Opt. Soc. Am. A 11, 1632–1640 (1994).
    [CrossRef]
  14. R. G. Dorsch, A. W. Lohmann, S. Sinzinger, “Fresnel ping-pong algorithm for 2-plane computer-generated hologram display,” Appl. Opt. 33, 869–875 (1994).
    [CrossRef] [PubMed]
  15. J. A. Cox, B. S. Fritz, T. R. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
    [CrossRef]
  16. T. H. P. Chang, “Proximity effect in electron-beam lithography,” J. Vac. Sci. Technol. 12, 1271–1275 (1975).
    [CrossRef]
  17. F. Nikolajeff, J. Bengtsson, M. Larsson, M. Ekberg, S. Hard, “Measuring and modeling the proximity effect in direct-write electron-beam lithography kinoforms,” Appl. Opt. 34, 897–903 (1995).
    [CrossRef] [PubMed]

2000 (1)

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

1997 (1)

1996 (2)

H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption,” Appl. Opt. 35, 6865–6874 (1996).
[CrossRef] [PubMed]

1995 (3)

1994 (4)

1993 (1)

1988 (1)

1983 (1)

1975 (1)

T. H. P. Chang, “Proximity effect in electron-beam lithography,” J. Vac. Sci. Technol. 12, 1271–1275 (1975).
[CrossRef]

1972 (1)

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

Aagedal, H.

H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

Bengtsson, J.

Beyerlein, M.

Bryngdahl, O.

Chang, M. P.

Chang, T. H. P.

T. H. P. Chang, “Proximity effect in electron-beam lithography,” J. Vac. Sci. Technol. 12, 1271–1275 (1975).
[CrossRef]

Cox, J. A.

J. A. Cox, B. S. Fritz, T. R. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
[CrossRef]

Dong, B. H.

Dong, B. Z.

Dorsch, R. G.

Dresel, T.

Duparre, M.

Ehbets, P.

Ekberg, M.

Ersoy, O. K.

Fritz, B. S.

J. A. Cox, B. S. Fritz, T. R. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
[CrossRef]

Gale, M. T.

Gerchberg, R. W.

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

Golub, M. A.

Gu, B. Y.

Han, C.-Y.

Hard, S.

Herzig, H. P.

Herzig, H.-P.

P. Ehbets, M. Rossi, H.-P. Herzig, “Continuous-relief fan-out elements with optimized fabrication tolerances,” Opt. Eng. 34, 3456–3464 (1995).
[CrossRef]

Ishii, Y.

Johansson, M.

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

Larsson, M.

Lohmann, A. W.

Ludge, B.

Murata, K.

Nikolajeff, F.

Pavelyev, V. S.

Prongue, D.

Rossi, M.

Saxton, W. O.

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

Schmid, M.

H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

Schütz, H.

Schwider, J.

Sinzinger, S.

Soifer, V. A.

Tan, X.

Teiwes, S.

H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

Uspleniev, G. V.

Volotovskii, S. G.

Werner, T. R.

J. A. Cox, B. S. Fritz, T. R. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
[CrossRef]

Wyrowski, F.

H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

Yang, G. Z.

Zhuang, J. Y.

Appl. Opt. (9)

C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[CrossRef] [PubMed]

M. Duparre, M. A. Golub, B. Ludge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2-laser beams,” Appl. Opt. 34, 2489–2497 (1995).
[CrossRef]

T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption,” Appl. Opt. 35, 6865–6874 (1996).
[CrossRef] [PubMed]

M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongue, “Continuous-relief diffractive optical elements for two-dimensional array generation,” Appl. Opt. 32, 2526–2533 (1993).
[CrossRef] [PubMed]

G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, O. K. Ersoy, “Gerchberg–Saxton and Yang–Gu algorithms for phase retrieval in a nonunitary transform system: a comparison,” Appl. Opt. 33, 209–218 (1994).
[CrossRef] [PubMed]

J. Bengtsson, “Design of fan-out kinoforms in the entire scalar diffraction regime with an optimal-rotation-angle method,” Appl. Opt. 36, 8435–8444 (1997).
[CrossRef]

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]

R. G. Dorsch, A. W. Lohmann, S. Sinzinger, “Fresnel ping-pong algorithm for 2-plane computer-generated hologram display,” Appl. Opt. 33, 869–875 (1994).
[CrossRef] [PubMed]

F. Nikolajeff, J. Bengtsson, M. Larsson, M. Ekberg, S. Hard, “Measuring and modeling the proximity effect in direct-write electron-beam lithography kinoforms,” Appl. Opt. 34, 897–903 (1995).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

M. Johansson, J. Bengtsson, “Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations,” J. Mod. Opt. 47, 1385–1398 (2000).
[CrossRef]

H. Aagedal, M. Schmid, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

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

J. Vac. Sci. Technol. (1)

T. H. P. Chang, “Proximity effect in electron-beam lithography,” J. Vac. Sci. Technol. 12, 1271–1275 (1975).
[CrossRef]

Opt. Eng. (1)

P. Ehbets, M. Rossi, H.-P. Herzig, “Continuous-relief fan-out elements with optimized fabrication tolerances,” Opt. Eng. 34, 3456–3464 (1995).
[CrossRef]

Optik (Stuttgart) (1)

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

Other (1)

J. A. Cox, B. S. Fritz, T. R. Werner, “Process error limitations on binary optics performance,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 80–88 (1991).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

(a) Cell in the DOE structure contributing to the field in the focus positions. (b) Complex plane construction used in the efficiency optimization. (c) Construction used in the minimization of the error sensitivity.

Fig. 2
Fig. 2

Block diagram of the design algorithm.

Fig. 3
Fig. 3

Design algorithm performance for the DOE giving a 2 × 2 fan-out: the error sensitivity in the focal positions as a function of the number of iterations. Also shown is how the weight factor, W, is changed.

Fig. 4
Fig. 4

Simulated sensitivity of the focus position intensities to a linear depth error for a DOE designed (a) with and (b) without sensitivity minimization. Inserts show a portion of the DOE relief as a gray scale in which black denotes the deepest positions.

Fig. 5
Fig. 5

Simulated uniformity error of the focus position intensities as a function of the linear depth error.

Fig. 6
Fig. 6

Measured sensitivity of the focus position intensities to a linear depth error for a fabricated DOE designed (a) with and (b) without sensitivity minimization.

Fig. 7
Fig. 7

Measured uniformity error of the focus position intensities as a function of the linear depth error for fabricated DOEs.

Fig. 8
Fig. 8

Design algorithm performance for the DOE giving a 2 × 2 fan-out and a zero-order fan-out.

Fig. 9
Fig. 9

Simulated sensitivity of the focus position intensities to a linear depth error for the DOE with a zero-order fan-out designed (a) with and (b) without sensitivity minimization. Inserts show a portion of the DOE relief.

Fig. 10
Fig. 10

Simulated uniformity error of the focus position intensities as a function of the linear depth error.

Fig. 11
Fig. 11

Measured sensitivity of the focus position intensities to a linear depth error for a fabricated DOE with zero-order fan-out designed (a) with and (b) without sensitivity minimization.

Fig. 12
Fig. 12

Measured uniformity error of the focus position intensities as a function of the linear depth error for fabricated DOEs with zero-order fan-out.

Fig. 13
Fig. 13

Photographs of the fan-out from the fabricated DOEs with five focus positions of which the central one corresponds to the zeroth diffraction order. Each fan-out is showed as an image of the exit slit of the monochromator. The different illuminating wavelengths are equivalent to linear depth errors in the surface relief: (a) λ = 660 nm, error ≈ 0%; (b) λ = 630 nm, error ≈ +5%; (c) λ = 600 nm, error ≈ +10%; (d) λ = 570 nm, error ≈ +16%; (e) λ = 530 nm, error ≈ +25%; and (f) λ = 500 nm, error ≈ +32%. The increased wavelength-independent behavior of the DOE that is designed with error-sensitivity minimization is clearly seen and indicates an increased tolerance to linear fabrication errors.

Equations (12)

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Ekm=Akm expjφkmexpjφkAkm expjαkm.
φk=2π/λ01-ndkfor transmission DOE4π/λ0dkfor reflection DOE,
ΔAm=Akm cosαm-αkm.
f1=m ΔAm.
f1=m wm ΔAm.
ΔAmδφk=Akm cosαm-αkm-δφk-Akm cosαm-αkm=Akm cosαm-αkmcosδφk+Akm sinαm-αkmsinδφk-Akm cosαm-αkmAkmδφk sinαm-αkm,
sm = k ΔAmδφk= constk Akmφk sinαm-αkm.
f2=m vmsm-s¯2,
s¯=1Mm sm,
ftotal=f1-Wf2.
Uerr=maxIm-minImmaxIm+minIm,
depthactual=1+linear errordepthdesign,

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