Abstract

Numerical simulations have been performed to study the influence of resist-developing errors on continuous-relief microlenses. For very low Fresnel numbers (N<4), the focal shift counteracts changes in the radius of curvature that are due to the depth errors and stabilizes the focal length. For higher Fresnel numbers (N>2), the focal length is essentially determined by the diffraction at the lateral pattern of the segments, and deviations from the ideal blaze influence only the efficiency. A qualitative picture based on Fourier optics is given to explain these markedly relaxed tolerances with respect to depth errors for planar optical elements and even for low Fresnel numbers.

© 1997 Optical Society of America

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References

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  1. R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).
  2. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  3. G. Y. Yoon, T. Jitsuno, M. Nakatsuka, S. Nakai, “Shack–Hartmann wave-front measurement with large F-number plastic microlens array,” Appl. Opt. 35, 188–192 (1996).
    [CrossRef] [PubMed]
  4. R. E. Kunz, M. Rossi, “Phase-matched Fresnel elements,” Opt. Commun. 97, 6–9 (1993).
    [CrossRef]
  5. E.-B. Kley, T. Possner, R. Göring, “Realization of micro-optics and integrated optic components by electron-beam-lithographic surface profiling and ion exchange in glass,” Int. J. Optoelectron. 8, 513–527 (1993).
  6. M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief microoptical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
    [CrossRef]
  7. C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
    [CrossRef]
  8. D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
    [CrossRef]
  9. M. Rossi, R. E. Kunz, H-P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
    [CrossRef] [PubMed]
  10. S. Sinzinger, M. Testorf, “Transition between diffractive and refractive micro-optical components,” Appl. Opt. 34, 5970–5976 (1995).
    [CrossRef] [PubMed]
  11. T. Hessler, M. T. Gale, M. Rossi, R. E. Kunz, “Fabrication of Fresnel microlens arrays by direct writing in photoresist,” in Vol. 5 of EOS Topical Meetings Digest Series (European Optical Society, Orsay, France, 1995), pp. 37–43.
  12. Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
    [CrossRef]
  13. J. H. Erkkila, “On the maximum wave intensity in the focal volume,” Opt. Commun. 43, 313–314 (1982).
    [CrossRef]
  14. J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 10.2.
  15. P. Ehbets, M. Rossi, H. P. Herzig, “Continuous-relief fan out elements with optimized fabrication tolerances,” Opt. Eng. 34, 3456–3464 (1995).
    [CrossRef]
  16. H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
    [CrossRef]
  17. M. Martinez-Corral, V. Climent, “Focal switch: a new effect in low-Fresnel-number systems,” Appl. Opt. 35, 24–27 (1996).
    [CrossRef]
  18. H. P. Herzig, “Design of refractive and diffractive micro-optics,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).
  19. M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 49.

1996 (3)

1995 (3)

1994 (1)

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

1993 (2)

R. E. Kunz, M. Rossi, “Phase-matched Fresnel elements,” Opt. Commun. 97, 6–9 (1993).
[CrossRef]

E.-B. Kley, T. Possner, R. Göring, “Realization of micro-optics and integrated optic components by electron-beam-lithographic surface profiling and ion exchange in glass,” Int. J. Optoelectron. 8, 513–527 (1993).

1990 (1)

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

1984 (1)

1982 (1)

J. H. Erkkila, “On the maximum wave intensity in the focal volume,” Opt. Commun. 43, 313–314 (1982).
[CrossRef]

1981 (1)

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

Blattner, P.

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

Blough, C. G.

C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
[CrossRef]

Climent, V.

Cullmann, E.

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

Daly, D.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Dändliker, R.

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

Davies, N.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Ehbets, P.

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

Erkkila, J. H.

J. H. Erkkila, “On the maximum wave intensity in the focal volume,” Opt. Commun. 43, 313–314 (1982).
[CrossRef]

Faklis, D.

C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
[CrossRef]

Gale, M. T.

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

T. Hessler, M. T. Gale, M. Rossi, R. E. Kunz, “Fabrication of Fresnel microlens arrays by direct writing in photoresist,” in Vol. 5 of EOS Topical Meetings Digest Series (European Optical Society, Orsay, France, 1995), pp. 37–43.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 10.2.

Göring, R.

E.-B. Kley, T. Possner, R. Göring, “Realization of micro-optics and integrated optic components by electron-beam-lithographic surface profiling and ion exchange in glass,” Int. J. Optoelectron. 8, 513–527 (1993).

Herzig, H. P.

H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

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

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

H. P. Herzig, “Design of refractive and diffractive micro-optics,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

Herzig, H-P.

Hessler, T.

T. Hessler, M. T. Gale, M. Rossi, R. E. Kunz, “Fabrication of Fresnel microlens arrays by direct writing in photoresist,” in Vol. 5 of EOS Topical Meetings Digest Series (European Optical Society, Orsay, France, 1995), pp. 37–43.

Hugle, W. B.

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

Hutley, M. C.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 49.

Jitsuno, T.

Kley, E.-B.

E.-B. Kley, T. Possner, R. Göring, “Realization of micro-optics and integrated optic components by electron-beam-lithographic surface profiling and ion exchange in glass,” Int. J. Optoelectron. 8, 513–527 (1993).

Kuittinen, M.

H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Kunz, R. E.

M. Rossi, R. E. Kunz, H-P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
[CrossRef] [PubMed]

R. E. Kunz, M. Rossi, “Phase-matched Fresnel elements,” Opt. Commun. 97, 6–9 (1993).
[CrossRef]

T. Hessler, M. T. Gale, M. Rossi, R. E. Kunz, “Fabrication of Fresnel microlens arrays by direct writing in photoresist,” in Vol. 5 of EOS Topical Meetings Digest Series (European Optical Society, Orsay, France, 1995), pp. 37–43.

Li, Y.

Mack, S. K.

C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
[CrossRef]

Martinez-Corral, M.

Michaels, R. L.

C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
[CrossRef]

Nakai, S.

Nakatsuka, M.

Nussbaum, Ph.

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

Pedersen, J.

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

Piper, E.

H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Possner, T.

E.-B. Kley, T. Possner, R. Göring, “Realization of micro-optics and integrated optic components by electron-beam-lithographic surface profiling and ion exchange in glass,” Int. J. Optoelectron. 8, 513–527 (1993).

Rossi, M.

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

M. Rossi, R. E. Kunz, H-P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
[CrossRef] [PubMed]

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

R. E. Kunz, M. Rossi, “Phase-matched Fresnel elements,” Opt. Commun. 97, 6–9 (1993).
[CrossRef]

T. Hessler, M. T. Gale, M. Rossi, R. E. Kunz, “Fabrication of Fresnel microlens arrays by direct writing in photoresist,” in Vol. 5 of EOS Topical Meetings Digest Series (European Optical Society, Orsay, France, 1995), pp. 37–43.

Schütz, H.

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

Singer, W.

H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Sinzinger, S.

Stevens, R. F.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Testorf, M.

Völkel, R.

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

Wangler, J.

H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Ward, S. J.

C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
[CrossRef]

Wolf, E.

Yoon, G. Y.

Appl. Opt. (4)

Int. J. Optoelectron. (1)

E.-B. Kley, T. Possner, R. Göring, “Realization of micro-optics and integrated optic components by electron-beam-lithographic surface profiling and ion exchange in glass,” Int. J. Optoelectron. 8, 513–527 (1993).

J. Meas. Sci. Technol. (1)

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photo-resist,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

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

Opt. Commun. (3)

R. E. Kunz, M. Rossi, “Phase-matched Fresnel elements,” Opt. Commun. 97, 6–9 (1993).
[CrossRef]

Y. Li, E. Wolf, “Focal shifts in diffracted converging spherical waves,” Opt. Commun. 39, 211–215 (1981).
[CrossRef]

J. H. Erkkila, “On the maximum wave intensity in the focal volume,” Opt. Commun. 43, 313–314 (1982).
[CrossRef]

Opt. Eng. (3)

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

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

H. P. Herzig, M. Kuittinen, W. Singer, J. Wangler, E. Piper, “Diffractive beam shaping elements at the fabrication limit,” Opt. Eng. 35, 2779–2787 (1996).
[CrossRef]

Other (6)

R. Völkel, H. P. Herzig, Ph. Nussbaum, P. Blattner, R. Dändliker, E. Cullmann, W. B. Hugle, “Microlens lithography and smart masks,” in Micro-Nano-Engineering 96, Microelectronic Engineering (Elsevier, Amsterdam, 1997).

H. P. Herzig, “Design of refractive and diffractive micro-optics,” in Micro-Optics: Elements, Systems, and Applications, H. P. Herzig, ed. (Taylor & Francis, London, 1997).

M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 49.

C. G. Blough, D. Faklis, S. K. Mack, R. L. Michaels, S. J. Ward, “High-efficiency replicated diffractive optics,” in Design, Fabrication, and Applications of Precision Plastic Optics, R. Hebert, ed., Proc. SPIE2600, 50–55 (1995).
[CrossRef]

T. Hessler, M. T. Gale, M. Rossi, R. E. Kunz, “Fabrication of Fresnel microlens arrays by direct writing in photoresist,” in Vol. 5 of EOS Topical Meetings Digest Series (European Optical Society, Orsay, France, 1995), pp. 37–43.

J. D. Gaskill, Linear Systems, Fourier Transforms and Optics (Wiley, New York, 1978), Chap. 10.2.

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Figures (10)

Fig. 1
Fig. 1

Surface-relief profile of (a) the diffractive lens and (b) the refractive lens investigated (f=40 mm). The surface profile of the depth-scaled lenses (depth scaling μ) is indicated by the dotted and the dashed curves.

Fig. 2
Fig. 2

Optical performance of depth-scaled lenses for high N: (a) diffractive, (b) refractive.

Fig. 3
Fig. 3

Influence of depth scaling on the focal length for different Fresnel numbers (of the unscaled lens) for a refractive design. The solid curves are obtained by evaluation of the maximum of Eq. (9), and the data denoted by the symbols are obtained by the numerical simulation method.

Fig. 4
Fig. 4

Normalized focal-length change resulting from the etching errors for a diffractive and a refractive design. For the diffractive lens, the stabilization of the focal length due to diffraction comes into play with the illumination of the second zone.

Fig. 5
Fig. 5

Influence of depth scaling on the focal length for different Fresnel numbers for a diffractive design.

Fig. 6
Fig. 6

Focal-length variation of depth-scaled (μ=1.25) diffractive lenses with different phase offsets φ0. The onset of the focal-length stabilization can be shifted to lower N with the phase offset φ0.

Fig. 7
Fig. 7

(a) On-axis intensity for a depth scaling of μ=1.0 and varying Fresnel number. (b) For a depth scaling of μ=1.25, the diffractive behavior evolves with increasing number of illuminated segments.

Fig. 8
Fig. 8

(a) Grating and blaze contributions for b=2 and a depth scaling of μ=1.25. The width wg of the peaks is already much narrower than the width wb of the envelope function gblaze. (b) Precursors of the diffraction orders have already formed in the resulting intensity distribution I(v).

Fig. 9
Fig. 9

(a) Grating and blaze contributions for a depth-scaled (μ=1.25) cylindrical diffractive lens with two segments illuminated. (b) On-axis intensity I(v) for the same lens.

Fig. 10
Fig. 10

(a) On-axis intensity pattern for a low-Fresnel-number depth-scaled diffractive lens (f=40 mm, N=8, μ=1.25). Results are compared for the Fourier optics approach in which Eq. (21) is used (solid curve), and for numerical calculations (dashed curve). (b) Difference ΔI(z) of the two intensity patterns.

Equations (25)

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

N=a2λf,
d(x)=d0+R-R2-x2d0-x22R,
d˜(x)=μd(x),
d˜(x)d˜0-x22R˜.
f˜=f/μ
I(uN)=I01-uN2πN2 sinc2(uN),
uN=N2 ΔzΔz+f
I0=πa2λf2.
I˜(u˜N)=I˜01-u˜N2πN˜2 sinc2(u˜N),
I˜0=πa2λf˜2,
u˜N=N˜2 Δz˜Δz˜+f˜,
u˜N(z, μ)=μN2 z-(f/μ)z.
I˜(z, μ)=fπNz2 sinc2N2 fz-μ.
u2(x, y)=-u1(xe, ye)z12iλr122exp(ikr12)dxedye.
Ψ(x)=[ϕ(x)+φ0]mod 2π,
t(x)=1Λ combxΛexpi2πμ xΛrectxΛrectxbΛ,
g(ν)=comb(νΛ)δν-μΛ[Λ sinc(νΛ)][bΛ sinc(νbΛ)],
g(ν)=bΛ2 sincν-μΛΛn=- sincν-nΛbΛ=bΛ2gblazeggrat,
t(x)=combx22λfexpi2πμ x22λfrectx22λfrectx22λf/b.
ξ=x22λf,
t(ξ)={comb(ξ)[exp(i2πμξ)rect(ξ)]}rect(ξ/b).
g(η)={comb(η)[δ(η-μ)sinc(η)]}b sinc(bη)
g(η)=b sinc(η-μ)n=- sinc[b(η-n)]=bgblazeggrat
I(z, μ, N)=|g(z, μ, N)|2=N22 sinc2fz-μn=- sinc2N2 fz-n
I(z, μ, N)=fπNz2 sinc2fz-μn=- sinc2N2 fz-n.

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