Abstract

We investigate the behavior of microlens arrays consisting of lenslets that have statistically distributed parameters (focal length, offset phase, lens center). In theoretical investigations we introduce a lens-array coherence parameter κ, which describes the statistical variations of the lenslets. In experiments we measured this parameter for concrete lens arrays.

© 1997 Optical Society of America

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References

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  1. W. Goltsos, M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392–1397 (1990).
    [CrossRef]
  2. M. E. Motamedi, A. P. Andrews, W. J. Gunning, “Micro-optic laser beam scanner,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 2–13 (1993).
    [CrossRef]
  3. E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665–2670 (1993).
    [CrossRef]
  4. R. Göring, W. Berner, E.-B. Kley, “Miniaturized optical systems for beam deflection and modulation,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 54–61, (1993).
    [CrossRef]
  5. M. F. Lewis, R. A. Wilson, “The use of lenslet arrays in spatial light modulators,” Pure Appl. Opt. 3, 143–150 (1994).
    [CrossRef]
  6. S. Glöckner, R. Göring, “Analysis of a micro-optical light modulator,” Appl. Opt. 36, 1467–1471 (1997).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 1, 13–14.
  8. M. C. Hutley, D. Daly, R. F. Stevens, “The Testing of Microlens Arrays,” in Proceedings of the IOP Short Meetings (National Physical Laboratory, Teddington, UK, 1991), Vol. 30, pp. 67–82.

1997 (1)

1994 (1)

M. F. Lewis, R. A. Wilson, “The use of lenslet arrays in spatial light modulators,” Pure Appl. Opt. 3, 143–150 (1994).
[CrossRef]

1993 (1)

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665–2670 (1993).
[CrossRef]

1990 (1)

W. Goltsos, M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392–1397 (1990).
[CrossRef]

Andrews, A. P.

M. E. Motamedi, A. P. Andrews, W. J. Gunning, “Micro-optic laser beam scanner,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 2–13 (1993).
[CrossRef]

Berner, W.

R. Göring, W. Berner, E.-B. Kley, “Miniaturized optical systems for beam deflection and modulation,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 54–61, (1993).
[CrossRef]

Daly, D.

M. C. Hutley, D. Daly, R. F. Stevens, “The Testing of Microlens Arrays,” in Proceedings of the IOP Short Meetings (National Physical Laboratory, Teddington, UK, 1991), Vol. 30, pp. 67–82.

Glöckner, S.

Goltsos, W.

W. Goltsos, M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392–1397 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 1, 13–14.

Göring, R.

S. Glöckner, R. Göring, “Analysis of a micro-optical light modulator,” Appl. Opt. 36, 1467–1471 (1997).
[CrossRef] [PubMed]

R. Göring, W. Berner, E.-B. Kley, “Miniaturized optical systems for beam deflection and modulation,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 54–61, (1993).
[CrossRef]

Gunning, W. J.

M. E. Motamedi, A. P. Andrews, W. J. Gunning, “Micro-optic laser beam scanner,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 2–13 (1993).
[CrossRef]

Holz, M.

W. Goltsos, M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392–1397 (1990).
[CrossRef]

Hutley, M. C.

M. C. Hutley, D. Daly, R. F. Stevens, “The Testing of Microlens Arrays,” in Proceedings of the IOP Short Meetings (National Physical Laboratory, Teddington, UK, 1991), Vol. 30, pp. 67–82.

Kley, E.-B.

R. Göring, W. Berner, E.-B. Kley, “Miniaturized optical systems for beam deflection and modulation,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 54–61, (1993).
[CrossRef]

Lewis, M. F.

M. F. Lewis, R. A. Wilson, “The use of lenslet arrays in spatial light modulators,” Pure Appl. Opt. 3, 143–150 (1994).
[CrossRef]

Motamedi, M. E.

M. E. Motamedi, A. P. Andrews, W. J. Gunning, “Micro-optic laser beam scanner,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 2–13 (1993).
[CrossRef]

Stevens, R. F.

M. C. Hutley, D. Daly, R. F. Stevens, “The Testing of Microlens Arrays,” in Proceedings of the IOP Short Meetings (National Physical Laboratory, Teddington, UK, 1991), Vol. 30, pp. 67–82.

Watson, E. A.

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665–2670 (1993).
[CrossRef]

Wilson, R. A.

M. F. Lewis, R. A. Wilson, “The use of lenslet arrays in spatial light modulators,” Pure Appl. Opt. 3, 143–150 (1994).
[CrossRef]

Appl. Opt. (1)

Opt. Eng. (2)

W. Goltsos, M. Holz, “Agile beam steering using binary optics microlens arrays,” Opt. Eng. 29, 1392–1397 (1990).
[CrossRef]

E. A. Watson, “Analysis of beam steering with decentered microlens arrays,” Opt. Eng. 32, 2665–2670 (1993).
[CrossRef]

Pure Appl. Opt. (1)

M. F. Lewis, R. A. Wilson, “The use of lenslet arrays in spatial light modulators,” Pure Appl. Opt. 3, 143–150 (1994).
[CrossRef]

Other (4)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 1, 13–14.

M. C. Hutley, D. Daly, R. F. Stevens, “The Testing of Microlens Arrays,” in Proceedings of the IOP Short Meetings (National Physical Laboratory, Teddington, UK, 1991), Vol. 30, pp. 67–82.

R. Göring, W. Berner, E.-B. Kley, “Miniaturized optical systems for beam deflection and modulation,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 54–61, (1993).
[CrossRef]

M. E. Motamedi, A. P. Andrews, W. J. Gunning, “Micro-optic laser beam scanner,” in Miniature and Micro-Optics and Micromechanics, N. C. Gallagher, C. S. Roychoudhuri, eds., Proc. SPIE1992, 2–13 (1993).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Microlens arrays in a confocal arrangement: (a) coherent array in which ideal arrays produce a wave front without statistical variations, (b) incoherent array in which inhomogeneous arrays result in statistical phase distributions.

Fig. 2
Fig. 2

Encircled energy plots for different lens-array coherence parameters κ (N = 20).

Fig. 3
Fig. 3

Encircled energy plots for different c and N = 20; the related lens-array coherence parameter κ is given in parentheses (c = 0.5/d corresponds to wave-front tilts with maximum deviations between ±0.25λ).

Fig. 4
Fig. 4

Encircled energy plots for different c (multiple of the wavelength of the defocusing error) and N = 20; the related lens-array coherence parameter κ is given in parentheses.

Fig. 5
Fig. 5

Tolerances to ensure coherent interaction of the lenslets in an array (λ = 670 nm): (a) focal-length variation, (b) lens pitch variation.

Fig. 6
Fig. 6

Misalignment errors for ideal arrays: (a) rotation about the optical axis; (b) tilt.

Fig. 7
Fig. 7

Test setup for microlens arrays: (a) with two arrays, (b) autocollimation setup.

Fig. 8
Fig. 8

Intensity distribution for array 1 (|fx| = |x′/λfL| ≤ 1/2d).

Fig. 9
Fig. 9

Measured encircled energy compared with the theoretical fit for array 1 (r0 = 1/2d).

Fig. 10
Fig. 10

Intensity distribution for array 2 (|fx| = |x′/λfL| ≤ 1/2d).

Fig. 11
Fig. 11

Measured encircled energy compared with theoretical fit for array 2 (r0 = 1/2d).

Fig. 12
Fig. 12

Surface profile of single lenses: (a) array 1, (b) array 2.

Fig. 13
Fig. 13

Section of the surface profile of lens array 1 showing the variation of the lens height.

Tables (3)

Tables Icon

Table 1 Lens–Array Coherence Parameter κ for Different Probability Distributions Ω

Tables Icon

Table 2 Tolerable Statistical Variations for Different Statistical Situations Assuming Uniform Probability Distributions and Corresponding κ

Tables Icon

Table 3 Comparison of the Results of Measurement and Surface Profilometry

Equations (29)

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uix-xi=Uifxexp-i2πfxxi,
uix=Uifx.
Uifx=Pˆfx, αiχfx.
Ifx=i=1N Pˆfx, αiχfxexp-i2πfxxi2
=i=1N Pˆfx, αiχfx2+2 i<ji,j=1N Pˆfx, αiχfxPˆfx, αjχfx*×exp-i2πfxxi-xj.
IfxΩ=i=1N Pˆfx, αiχfx2Ω+2 i<ji,j=1NPˆfx, αiχfxPˆfx, αjχfx*×exp-i2πfxxi-xjΩ.
IfxΩ=N -Pˆfx, αχfx2Ωαdα+2- Pˆfx, αχfxΩαdα2×i<ji,j=1N cos2πfxxi-xj.
2  i<ji,j=1N cos2πfxxi-xj=d2N2 sinc2dNfxd2 sinc2dfx-N=NχNfxχfx2-N.
IfxΩ=N - Pˆfx, αχfx2Ωαdα+- Pˆfx, αχfxΩαdα2×NχNfxχfx2-N.
Ωα=δα-α0.
Ifx=Pˆfx, α0χfx2 NχNfxχfx2.
Pˆfx, αχfx=expiαχfx.
κ=- expiαΩαdα2.
IfxΩ=κIcohfx+1-κ Iincohfx.
vr=-rr IfxΩdfx-1/2d1/2d IfxΩdfx.
Pˆfx, αχfx=χfx-α.
IfxΩ=N - χ2fx-αΩαdα+- χfx-αΩαdα2NχNfxχfx2-N.
Ωα=12c rectα2c.
Pˆfx, αχfx=expi2πα2xd2rectxd=expi2πα2xd2χfx.
Ωα=12c rectα2c.
ϕr=πr2λf Δff.
Δf<2λf#2.
Δx<λf#4.
Δβλf#22D.
Δγdefocus4λf#2D,
Δγtilt4λf#D1/2.
rectx=1:x1/20:x>1/2,
sincx=sinπxπx,
rectx=sincfx.

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