Abstract

This paper deals with a computer simulation and an experimental realization of an optical setup for automatic quality control of microlens arrays. The method is based on a 4f coherent light correlator setup with an amplitude filter placed in the Fourier plane. The output intensity signal is proportional to the first derivative of the distortion of the input wavefront. An analysis can be carried out with the use of the Zernike polynomial expansion method. It must be carried out separately for each lens, but it allows for a more precise, quantitative assessment of their quality. What is important is that the analysis is computer-based and performed on the basis of the initial single optical measurement.

© 2010 Optical Society of America

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  1. L. Davidson and R. Keller, “Basics of a light microscopy imaging system and its application in biology,” in Methods in Cellular Imaging, A.Periasamy, ed. (Oxford, 2001), pp. 53–65.
  2. H. J. Tiziani, T. Haist, and S. Reuter, “Optical inspection and characterization of micro-optics using confocal microscopy,” Opt. Laser Eng. 36, 403–415 (2001).
    [CrossRef]
  3. D. Murphy, “Phase contrast microscopy,” in Fundamentals of Light Microscopy and Electronic Imaging (Wiley, 2001).
  4. S. Reichelt and H. Zappe, “Combined Twyman–Green and Mach–Zehnder interferometer for microlens testing,” Appl. Opt. 44, 5786–5792 (2005).
    [CrossRef] [PubMed]
  5. S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
    [CrossRef]
  6. K. Kincaid, “Optical surface analyzers become precision manufacturing tools,” Laser Focus World 41, 89–93 (2005).
  7. S. C. West, "Interferometric Hartmann wave-front sensing for active optics at the 6.5-m conversion of the Multiple Mirror Telescope," Appl. Opt. 41, 3781–3789 (2002).
    [CrossRef] [PubMed]
  8. M. Davidson and M. Abramowitz, Encyclopedia of Imaging Science and Technology, J.Hornak, ed. (Wiley, 2002), Vol. II.
  9. E. Slayter and H. Slayter, “Imaging of phase objects,” in Light and Electron Microscopy, E.Slayter and H.Slayter, eds. (Cambridge, 1992), pp. 149–167.
  10. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef] [PubMed]
  11. G. Settles, Schlieren and Shadowgraph Techniques (Springer–Verlag, 2001).
    [CrossRef]
  12. L. Joannes, F. Dubois, and J. Legros, “Phase-shifting schlieren: High-resolution quantitative schlieren that uses the phase-shifting technique principle,” Appl. Opt. 42, 5046–5053(2003).
    [CrossRef] [PubMed]
  13. B. Zakharin and J. Stricker, “Schlieren systems with coherent illumination for quantitative measurements,” Appl. Opt. 43, 4786–4795 (2004).
    [CrossRef] [PubMed]
  14. T. Poon and K. B. Doh, “On the theory of optical Hilbert transform for incoherent objects,” Opt. Express 15, 3006–3011(2007).
    [CrossRef] [PubMed]
  15. R. Hoffman and L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
    [CrossRef] [PubMed]
  16. B. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181–186 (1978).
    [CrossRef] [PubMed]
  17. H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
    [CrossRef] [PubMed]
  18. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M.Fernández-Alonso, “Fractional derivative Fourier plane filter for phase change visualization,” Appl. Opt. 36, 7461–7464(1997).
    [CrossRef]
  19. E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernández-Alonso, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490(1997).
    [CrossRef]
  20. T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, and M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998).
    [CrossRef]
  21. A. Sagan, S. Nowicki, R. Buczyński, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816–5824 (2003).
    [CrossRef] [PubMed]
  22. R. Kasztelanic and A. Sagan, “Semiderivative real filter for micro-optical element quality control,” Opt. Rev. 16, 252–256(2009).
    [CrossRef]
  23. W. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. 70, 998–1005 (1980).
    [CrossRef]
  24. T. M. Jeong, D. Ko, and J. Lee, "Method of reconstructing wavefront aberrations by use of Zernike polynomials in radial shearing interferometers," Opt. Lett. 32, 232–234 (2007).
    [CrossRef] [PubMed]
  25. W. Zou and J. Rolland, “Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils,” J. Opt. Soc. Am. A 22, 938–951 (2005).
    [CrossRef]
  26. K. Freischlad and C. Koliopoulos, “Modal estimation of a wavefront from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861(1986).
    [CrossRef]
  27. F. Roddier and C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327(1991).
    [CrossRef] [PubMed]
  28. J. Hermann, “Least-squares wavefront errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
    [CrossRef]
  29. S. Dingqiang, J. Shengtao, and S. Lianzhen, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Progress in Telescope and Instrumentation Technologies, M.H.Ulrich, eds. (European Southern Observatory, 1992), pp. 289–292.

2009 (1)

R. Kasztelanic and A. Sagan, “Semiderivative real filter for micro-optical element quality control,” Opt. Rev. 16, 252–256(2009).
[CrossRef]

2007 (2)

2005 (4)

S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
[CrossRef]

K. Kincaid, “Optical surface analyzers become precision manufacturing tools,” Laser Focus World 41, 89–93 (2005).

W. Zou and J. Rolland, “Iterative zonal wave-front estimation algorithm for optical testing with general-shaped pupils,” J. Opt. Soc. Am. A 22, 938–951 (2005).
[CrossRef]

S. Reichelt and H. Zappe, “Combined Twyman–Green and Mach–Zehnder interferometer for microlens testing,” Appl. Opt. 44, 5786–5792 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (2)

2002 (1)

2001 (1)

H. J. Tiziani, T. Haist, and S. Reuter, “Optical inspection and characterization of micro-optics using confocal microscopy,” Opt. Laser Eng. 36, 403–415 (2001).
[CrossRef]

1998 (1)

1997 (2)

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M.Fernández-Alonso, “Fractional derivative Fourier plane filter for phase change visualization,” Appl. Opt. 36, 7461–7464(1997).
[CrossRef]

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernández-Alonso, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490(1997).
[CrossRef]

1991 (1)

1986 (1)

K. Freischlad and C. Koliopoulos, “Modal estimation of a wavefront from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861(1986).
[CrossRef]

1982 (1)

1980 (2)

1978 (1)

1975 (1)

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Aatz, B.

S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
[CrossRef]

Abramowitz, M.

M. Davidson and M. Abramowitz, Encyclopedia of Imaging Science and Technology, J.Hornak, ed. (Wiley, 2002), Vol. II.

Bieber, A.

S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
[CrossRef]

Buczynski, R.

Climent, V.

Davidson, L.

L. Davidson and R. Keller, “Basics of a light microscopy imaging system and its application in biology,” in Methods in Cellular Imaging, A.Periasamy, ed. (Oxford, 2001), pp. 53–65.

Davidson, M.

M. Davidson and M. Abramowitz, Encyclopedia of Imaging Science and Technology, J.Hornak, ed. (Wiley, 2002), Vol. II.

Dingqiang, S.

S. Dingqiang, J. Shengtao, and S. Lianzhen, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Progress in Telescope and Instrumentation Technologies, M.H.Ulrich, eds. (European Southern Observatory, 1992), pp. 289–292.

Doh, K. B.

Dubois, F.

Fernández-Alonso, M.

Freischlad, K.

K. Freischlad and C. Koliopoulos, “Modal estimation of a wavefront from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861(1986).
[CrossRef]

Gross, L.

Haist, T.

H. J. Tiziani, T. Haist, and S. Reuter, “Optical inspection and characterization of micro-optics using confocal microscopy,” Opt. Laser Eng. 36, 403–415 (2001).
[CrossRef]

Hermann, J.

Hoffman, R.

Horwitz, B.

Jeong, T. M.

Joannes, L.

Kasprzak, H.

Kasztelanic, R.

R. Kasztelanic and A. Sagan, “Semiderivative real filter for micro-optical element quality control,” Opt. Rev. 16, 252–256(2009).
[CrossRef]

Keller, R.

L. Davidson and R. Keller, “Basics of a light microscopy imaging system and its application in biology,” in Methods in Cellular Imaging, A.Periasamy, ed. (Oxford, 2001), pp. 53–65.

Kincaid, K.

K. Kincaid, “Optical surface analyzers become precision manufacturing tools,” Laser Focus World 41, 89–93 (2005).

Ko, D.

Koliopoulos, C.

K. Freischlad and C. Koliopoulos, “Modal estimation of a wavefront from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861(1986).
[CrossRef]

Kowalczyk, M.

Lancis, J.

Lee, J.

Legros, J.

Lianzhen, S.

S. Dingqiang, J. Shengtao, and S. Lianzhen, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Progress in Telescope and Instrumentation Technologies, M.H.Ulrich, eds. (European Southern Observatory, 1992), pp. 289–292.

Murphy, D.

D. Murphy, “Phase contrast microscopy,” in Fundamentals of Light Microscopy and Electronic Imaging (Wiley, 2001).

Nowicki, S.

Poon, T.

Reichelt, S.

S. Reichelt and H. Zappe, “Combined Twyman–Green and Mach–Zehnder interferometer for microlens testing,” Appl. Opt. 44, 5786–5792 (2005).
[CrossRef] [PubMed]

S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
[CrossRef]

Reuter, S.

H. J. Tiziani, T. Haist, and S. Reuter, “Optical inspection and characterization of micro-optics using confocal microscopy,” Opt. Laser Eng. 36, 403–415 (2001).
[CrossRef]

Roddier, C.

Roddier, F.

Rolland, J.

Sagan, A.

Settles, G.

G. Settles, Schlieren and Shadowgraph Techniques (Springer–Verlag, 2001).
[CrossRef]

Shengtao, J.

S. Dingqiang, J. Shengtao, and S. Lianzhen, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Progress in Telescope and Instrumentation Technologies, M.H.Ulrich, eds. (European Southern Observatory, 1992), pp. 289–292.

Slayter, E.

E. Slayter and H. Slayter, “Imaging of phase objects,” in Light and Electron Microscopy, E.Slayter and H.Slayter, eds. (Cambridge, 1992), pp. 149–167.

Slayter, H.

E. Slayter and H. Slayter, “Imaging of phase objects,” in Light and Electron Microscopy, E.Slayter and H.Slayter, eds. (Cambridge, 1992), pp. 149–167.

Southwell, W.

Stricker, J.

Szoplik, T.

Tajahuerce, E.

Tiziani, H. J.

H. J. Tiziani, T. Haist, and S. Reuter, “Optical inspection and characterization of micro-optics using confocal microscopy,” Opt. Laser Eng. 36, 403–415 (2001).
[CrossRef]

West, S. C.

Zakharin, B.

Zappe, H.

S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
[CrossRef]

S. Reichelt and H. Zappe, “Combined Twyman–Green and Mach–Zehnder interferometer for microlens testing,” Appl. Opt. 44, 5786–5792 (2005).
[CrossRef] [PubMed]

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Zou, W.

Appl. Opt. (11)

R. Hoffman and L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
[CrossRef] [PubMed]

B. Horwitz, “Phase image differentiation with linear intensity output,” Appl. Opt. 17, 181–186 (1978).
[CrossRef] [PubMed]

H. Kasprzak, “Differentiation of a noninteger order and its optical implementation,” Appl. Opt. 21, 3287–3291 (1982).
[CrossRef] [PubMed]

J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, and M.Fernández-Alonso, “Fractional derivative Fourier plane filter for phase change visualization,” Appl. Opt. 36, 7461–7464(1997).
[CrossRef]

T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, and M. Fernández-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998).
[CrossRef]

S. C. West, "Interferometric Hartmann wave-front sensing for active optics at the 6.5-m conversion of the Multiple Mirror Telescope," Appl. Opt. 41, 3781–3789 (2002).
[CrossRef] [PubMed]

L. Joannes, F. Dubois, and J. Legros, “Phase-shifting schlieren: High-resolution quantitative schlieren that uses the phase-shifting technique principle,” Appl. Opt. 42, 5046–5053(2003).
[CrossRef] [PubMed]

A. Sagan, S. Nowicki, R. Buczyński, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with square-root, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816–5824 (2003).
[CrossRef] [PubMed]

B. Zakharin and J. Stricker, “Schlieren systems with coherent illumination for quantitative measurements,” Appl. Opt. 43, 4786–4795 (2004).
[CrossRef] [PubMed]

S. Reichelt and H. Zappe, “Combined Twyman–Green and Mach–Zehnder interferometer for microlens testing,” Appl. Opt. 44, 5786–5792 (2005).
[CrossRef] [PubMed]

F. Roddier and C. Roddier, “Wavefront reconstruction using iterative Fourier transforms,” Appl. Opt. 30, 1325–1327(1991).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Hermann, “Least-squares wavefront errors of minimum norm,” J. Opt. Soc. Am. 70, 28–35 (1980).
[CrossRef]

W. Southwell, “Wavefront estimation from wavefront slope measurements,” J. Opt. Soc. Am. 70, 998–1005 (1980).
[CrossRef]

K. Freischlad and C. Koliopoulos, “Modal estimation of a wavefront from difference measurements using the discrete Fourier transform,” J. Opt. Soc. Am. 3, 1852–1861(1986).
[CrossRef]

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

Laser Focus World (1)

K. Kincaid, “Optical surface analyzers become precision manufacturing tools,” Laser Focus World 41, 89–93 (2005).

Opt. Express (1)

Opt. Laser Eng. (1)

H. J. Tiziani, T. Haist, and S. Reuter, “Optical inspection and characterization of micro-optics using confocal microscopy,” Opt. Laser Eng. 36, 403–415 (2001).
[CrossRef]

Opt. Lett. (1)

Opt. Rev. (1)

R. Kasztelanic and A. Sagan, “Semiderivative real filter for micro-optical element quality control,” Opt. Rev. 16, 252–256(2009).
[CrossRef]

Proc. SPIE (1)

S. Reichelt, A. Bieber, B. Aatz, and H. Zappe, “Micro-optics metrology using advanced interferometry,” Proc. SPIE 5856, 437–446 (2005).
[CrossRef]

Pure Appl. Opt. (1)

E. Tajahuerce, T. Szoplik, J. Lancis, V. Climent, and M. Fernández-Alonso, “Phase-object fractional differentiation using Fourier plane filters,” Pure Appl. Opt. 6, 481–490(1997).
[CrossRef]

Science (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[CrossRef] [PubMed]

Other (6)

G. Settles, Schlieren and Shadowgraph Techniques (Springer–Verlag, 2001).
[CrossRef]

L. Davidson and R. Keller, “Basics of a light microscopy imaging system and its application in biology,” in Methods in Cellular Imaging, A.Periasamy, ed. (Oxford, 2001), pp. 53–65.

D. Murphy, “Phase contrast microscopy,” in Fundamentals of Light Microscopy and Electronic Imaging (Wiley, 2001).

M. Davidson and M. Abramowitz, Encyclopedia of Imaging Science and Technology, J.Hornak, ed. (Wiley, 2002), Vol. II.

E. Slayter and H. Slayter, “Imaging of phase objects,” in Light and Electron Microscopy, E.Slayter and H.Slayter, eds. (Cambridge, 1992), pp. 149–167.

S. Dingqiang, J. Shengtao, and S. Lianzhen, “A sort of algorithm of wavefront reconstruction for Shack–Hartmann test,” in Progress in Telescope and Instrumentation Technologies, M.H.Ulrich, eds. (European Southern Observatory, 1992), pp. 289–292.

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Figures (7)

Fig. 1
Fig. 1

Scheme of the setup for measurement of the quality of the microlens arrays.

Fig. 2
Fig. 2

Cross section of the amplitude real filter. (a) Linearly graded filter. (b) Semiderivative filter. W is the total width of the filter.

Fig. 3
Fig. 3

Zernike polynomial expansion for distortion added to the particular lenses. The shades of gray denote the consecutive terms of Zernike polynomial expansion.

Fig. 4
Fig. 4

E error in the reconstruction of the shape of the parabolic lens (a) and σ 2 error of Zernike polynomial expansion (b) with regard to the kind of filter for the camera with an 8–10 dynamic range.

Fig. 5
Fig. 5

Results of the reconstruction of variously shaped lenses by linear filter. (a) Error made during the reconstruction. (b) Values of Zernike expansion coefficients.

Fig. 6
Fig. 6

Zernike reconstruction of the shape of the microlens arrays. (a) Image registered on the CCD camera for the horizontal linear filter. (c) Automatically located centers and areas of the individual lenses. (b) Example of Zernike reconstruction of nine microlenses. (d) Exemplary Zernike expansion coefficients.

Fig. 7
Fig. 7

Zernike reconstruction of the shape of the microlenses (all dimensions given in μm ).

Equations (17)

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

t ( x , y ) = exp [ i θ ( x , y ) ] .
t ( u , v ) = { 0 u < w 2 a + u w w 2 u w 2 1 u > w 2 ,
I ( x , y ) = A 0 2 [ a + λ f 2 π w θ ( x , y ) x ] 2 ,
t ( u , v ) = { 0 u < w 2 a + u w w 2 u w 2 1 u > w 2 ,
I ( x , y ) = A 0 2 [ a + λ f 2 π w θ ( x , y ) x ] .
W ( ρ , θ ) = m n C n m Z n m ( ρ , θ ) ,
{ Z n m ( ρ , θ ) Z n m ( ρ , θ ) } = R n m ( ρ ) { sin ( m θ ) cos ( m θ ) } ,
R n m ( ρ ) = k = 0 ( n m ) / 2 ( 1 ) k ( n k ) ! k ! [ n + m 2 k ] ! [ n m 2 k ] ! ρ n 2 k n = 0 , 1 , 2 , ( n m ) even .
θ x = k = 0 M a k Z k ( x , y ) x , θ y = k = 0 M a k Z k ( x , y ) y ,
S = A a ,
a = ( A T A ) 1 A T S .
h ( r ) = h 0 R r 2 1 + 1 ( K + 1 ) r 2 / R 2 ,
θ ( r ) = h ( r ) 2 π ( n 1 ) λ .
θ ( r ) r | max r = R / 2 = h 0 2 π ( n 1 ) λ 3 k .
W = 2 f h 0 ( n 1 ) 3 k .
σ 2 = k = 0 M ( a k c k ) 2 ,
E = S ( H S h S ) 2 ,

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