Abstract

We present a method to obtain the average lenslet shape of microlens arrays and especially their aberrations from the far-field intensity distribution of the whole array. The method is based on the phase-retrieval algorithm introduced by Gerchberg and Saxton [Optik (Stuttgart) 35, 237 (1972)]. We show how to overcome the crucial point of this algorithm, that is finding suitable start parameters to end up with correct results. The procedure is successfully applied to a cylindrical microlens array produced by reflow technique and the result is compared with surface profilometric measurements. The technique is applicable for lenslets having small numerical apertures and fill factors near unity.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Streibl, U. Nölscher, J. Jahns, S. Walker, “Array generation with lenslet arrays,” Appl. Opt. 30, 2739–2742 (1991).
    [CrossRef] [PubMed]
  2. H. Herzig, Micro-Optics (Taylor & Francis, London, 1997).
  3. S. Sinzinger, J. Jahns, Microoptics (Wiley, New York, 1999).
  4. C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate,” J. Opt. A: Pure Appl. Opt. 1, 398–403 (1999).
    [CrossRef]
  5. P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).
  6. A. Büttner, U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. (to be published).
  7. D. Malacara, Optical Shop Testing (Wiley, New York, 1991).
  8. L. G. Hale, E. M. Motamedi, W. J. Gunning, “Optical testing and characterization of microlens arrays,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. S. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 47–51 (1993).
  9. J. Schwider, H. Sickinger, “Arraytests for microlenses,” Optik (Stuttgart) 107, 26–34 (1997).
  10. S. Glöckner, R. Göring, “Investigation of statistical variations between lenslets of microlens arrays,” Appl. Opt. 36, 4438–4445 (1997).
    [CrossRef] [PubMed]
  11. R. Gerchberg, W. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  13. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  14. H. Aagedal, M. Schmid, T. Beth, S. Wyrowski, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
    [CrossRef]
  15. B. Kress, P. Meyrueis, Digital Diffractive Optics (Wiley, New York, 2000).
  16. A. V. Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
    [CrossRef]
  17. A. V. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).
  18. J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).
  19. J. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” J. Opt. Soc. Am. A 4, 118–123 (1987).
    [CrossRef]
  20. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [CrossRef]
  21. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  22. V. Arrizón, M. Testorf, S. Sinzinger, J. Jahns, “Iterative optimization of phase-only diffractive optical elements based on a lenslet array,” J. Opt. Soc. Am. A 17, 2157–2164 (2000).
    [CrossRef]
  23. Z. Popovic, R. Sprague, G. Connell, “Technique for monolithic fabrication of microlens arrays,” Appl. Opt. 27, 1281–1284 (1988).
    [CrossRef] [PubMed]
  24. F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]

2000

1999

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate,” J. Opt. A: Pure Appl. Opt. 1, 398–403 (1999).
[CrossRef]

1997

1996

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

1991

1988

1987

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1978

F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

1972

R. Gerchberg, W. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

1963

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Aagedal, H.

A. V. Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
[CrossRef]

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

Arrizón, V.

Beth, T.

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

Büttner, A.

A. Büttner, U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. (to be published).

Connell, G.

Drauschke, A.

Eisner, M.

P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).

Fienup, J.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Gerchberg, R.

R. Gerchberg, W. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

Glöckner, S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Göring, R.

Gunning, W. J.

L. G. Hale, E. M. Motamedi, W. J. Gunning, “Optical testing and characterization of microlens arrays,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. S. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 47–51 (1993).

Hale, L. G.

L. G. Hale, E. M. Motamedi, W. J. Gunning, “Optical testing and characterization of microlens arrays,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. S. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 47–51 (1993).

Harris, F.

F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Haselbeck, S.

P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).

Herzig, H.

P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).

H. Herzig, Micro-Optics (Taylor & Francis, London, 1997).

Jahns, J.

Kopp, C.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate,” J. Opt. A: Pure Appl. Opt. 1, 398–403 (1999).
[CrossRef]

Kress, B.

B. Kress, P. Meyrueis, Digital Diffractive Optics (Wiley, New York, 2000).

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1991).

Meyrueis, P.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate,” J. Opt. A: Pure Appl. Opt. 1, 398–403 (1999).
[CrossRef]

B. Kress, P. Meyrueis, Digital Diffractive Optics (Wiley, New York, 2000).

Motamedi, E. M.

L. G. Hale, E. M. Motamedi, W. J. Gunning, “Optical testing and characterization of microlens arrays,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. S. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 47–51 (1993).

Nölscher, U.

Nussbaum, P.

P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Pfeil, A. V.

A. V. Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
[CrossRef]

A. V. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

Popovic, Z.

Ravel, L.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate,” J. Opt. A: Pure Appl. Opt. 1, 398–403 (1999).
[CrossRef]

Saxton, W.

R. Gerchberg, W. 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, T. Beth, S. Wyrowski, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

Schwider, J.

J. Schwider, H. Sickinger, “Arraytests for microlenses,” Optik (Stuttgart) 107, 26–34 (1997).

Sickinger, H.

J. Schwider, H. Sickinger, “Arraytests for microlenses,” Optik (Stuttgart) 107, 26–34 (1997).

Sinzinger, S.

Sprague, R.

Streibl, N.

Testorf, M.

Turunen, J.

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).

Völkel, R.

P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).

Walker, S.

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Wyrowski, F.

A. V. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

A. V. Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
[CrossRef]

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

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).

Wyrowski, S.

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

Zeitner, U. D.

A. Büttner, U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. (to be published).

Appl. Opt.

J. Mod. Opt.

A. V. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

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

J. Opt. A: Pure Appl. Opt.

C. Kopp, L. Ravel, P. Meyrueis, “Efficient beamshaper homogenizer design combining diffractive optical elements, microlens array, and random phase plate,” J. Opt. A: Pure Appl. Opt. 1, 398–403 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Acta

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[CrossRef]

Opt. Eng.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Optik (Stuttgart)

R. Gerchberg, W. Saxton, “A Practical Algorithm for the Determination of Phase from Image and Diffraction Plane Pictures,” Optik (Stuttgart) 35, 237–246 (1972).

J. Schwider, H. Sickinger, “Arraytests for microlenses,” Optik (Stuttgart) 107, 26–34 (1997).

Proc. IEEE

F. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

P. Nussbaum, R. Völkel, H. Herzig, M. Eisner, S. Haselbeck, “Design, fabrication, and testing of micro-optical components for sensors and microsystems,” in Micro-Optical Technologies for Measurement, Sensors and Microsystems II and Optical Fiber Sensor Technologies and Applications, O. M. Parriaux, B. Culshaw, M. Breidne, E.-B. Kiley, eds., Proc. SPIE3099, 196–211 (1997).

A. Büttner, U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. (to be published).

D. Malacara, Optical Shop Testing (Wiley, New York, 1991).

L. G. Hale, E. M. Motamedi, W. J. Gunning, “Optical testing and characterization of microlens arrays,” in Miniature and Micro-Optics: Fabrication and System Applications II, C. S. Roychoudhuri, W. B. Veldkamp, eds., Proc. SPIE1751, 47–51 (1993).

J. Turunen, F. Wyrowski, Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).

B. Kress, P. Meyrueis, Digital Diffractive Optics (Wiley, New York, 2000).

H. Herzig, Micro-Optics (Taylor & Francis, London, 1997).

S. Sinzinger, J. Jahns, Microoptics (Wiley, New York, 1999).

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 (9)

Fig. 1
Fig. 1

Optical setup to obtain the far-field intensity distribution of a microlens array.

Fig. 2
Fig. 2

Separation of the far field of a single lenslet.

Fig. 3
Fig. 3

Determination of the parabolic phase distribution used as an initial phase distribution for the Gerchberg and Saxton algorithm.

Fig. 4
Fig. 4

Scheme of the Gerchberg and Saxton algorithm.

Fig. 5
Fig. 5

Experimental setup for measuring the far-field intensity distribution of a microlens array.

Fig. 6
Fig. 6

Experimental demonstration of the far-field method (λ = 635 nm). (a) Measured far-field intensity distribution caused by approximately 50 lenslets. (b) Average far-field amplitude distribution caused by a single lenslet, filtered from (a) by applying a Gaussian amplitude filter. The solid curve in (b) is the retrieved far-field amplitude distribution after 10,000 iterations of the GSA, (c) is the calculated deviation from the corresponding best-SNR parabolic lenslet shape. (d) Deviation of the resulting shape error from the best-fitted parabolic lenslet shape, obtained by subtracting the second-order polynomial fit from the resulting shape error in (c).

Fig. 7
Fig. 7

Comparison of the fourth-order polynomial fit of the retrieved wavefront error displayed in Fig. 6(d) with the wavefront error of one lenslet of the array under test resulting from a surface profilometric measurement (λ = 635 nm).

Fig. 8
Fig. 8

Influence of fill factors smaller than unity on the resulting lens shape for an array consisting of 10 identical lenslets with p = 400 µm and f = 2000 µm. The curve belonging to the left y axis (straight curve) shows the rms deviation from the best-fitting parabolic lens, the plot belonging to the right y axis (dashed curve) shows the deviation of the focal length of the best-fitting parabolic lenses from the focal length of the predicted parabolic lens f = 2000 µm.

Fig. 9
Fig. 9

Influence of statistical lenslet variations in focal length and aberration on the retrieved lens shape. The graph shows the rms deviation of the calculated average lenslet shape from the predicted average lenslet shape. Δf max denotes the maximum focal length deviation of each curve.

Tables (1)

Tables Icon

Table 1 Properties of the Cylindrical Microlens Array under Test

Equations (11)

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

rectx1/p=1:|x1|12p0:|x1|>12p
UAx1, z1=rectx1Npcombx1p  ULx1,
combx1p=l=- δx1-lp.
|UMx2, z2|2=|UAx1, z1|2=|SNx2ULx1, z1|2.
|UFx2, z2|2=|ULP||UMx2, z2|2,
hx1=λ2πn-1Φx1.
ΔΦx1=ΦPx1-Φx1
dx1=λfFMdx2,
dx1max=λfFX2min=λ4NA.
Mmin=X2mindx2max.
|UFx2, z2|2=-1|ULP|  |UMx2, z2|2.

Metrics