Abstract

This Letter presents a theoretical and experimental image formation study in the presence of astigmatic aberrations. A three-dimensional, macroscopic location scheme of micrometer-sized particles for the single camera astigmatism particle tracking velocimetry (APTV) technique is introduced. Average particle z position determination errors of the technique are as low as 0.33%, with a measurement depth of 40 mm. These accuracies show APTV’s ability of measuring volumetric velocity fields in macroscopic domains with limited optical access.

© 2014 Optical Society of America

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References

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  1. C. Cierpka and C. J. Kähler, J. Visualization 15, 1 (2012).
    [CrossRef]
  2. H. Pin Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
    [CrossRef]
  3. C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
    [CrossRef]
  4. C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
    [CrossRef]
  5. F. Scarano, Meas. Sci. Technol. 24, 012001 (2013).
    [CrossRef]
  6. C. E. Towers and D. P. Towers, Opt. Lett. 31, 1220 (2006).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 397/460/478.
  8. N. G. van Kampen, Physica 14, 575 (1949).
    [CrossRef]
  9. B. R. A. Nijboer, Physica 13, 605 (1947).
    [CrossRef]
  10. K. Nienhuis and B. R. A. Nijboer, Physica 14, 590 (1949).
    [CrossRef]
  11. N. G. van Kampen, Physica 16, 817 (1950).
    [CrossRef]

2013

F. Scarano, Meas. Sci. Technol. 24, 012001 (2013).
[CrossRef]

2012

C. Cierpka and C. J. Kähler, J. Visualization 15, 1 (2012).
[CrossRef]

2011

C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
[CrossRef]

2010

C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
[CrossRef]

2006

1994

H. Pin Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

1950

N. G. van Kampen, Physica 16, 817 (1950).
[CrossRef]

1949

K. Nienhuis and B. R. A. Nijboer, Physica 14, 590 (1949).
[CrossRef]

N. G. van Kampen, Physica 14, 575 (1949).
[CrossRef]

1947

B. R. A. Nijboer, Physica 13, 605 (1947).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 397/460/478.

Cierpka, C.

C. Cierpka and C. J. Kähler, J. Visualization 15, 1 (2012).
[CrossRef]

C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
[CrossRef]

C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
[CrossRef]

Hain, R.

C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
[CrossRef]

Kähler, C. J.

C. Cierpka and C. J. Kähler, J. Visualization 15, 1 (2012).
[CrossRef]

C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
[CrossRef]

C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
[CrossRef]

Nienhuis, K.

K. Nienhuis and B. R. A. Nijboer, Physica 14, 590 (1949).
[CrossRef]

Nijboer, B. R. A.

K. Nienhuis and B. R. A. Nijboer, Physica 14, 590 (1949).
[CrossRef]

B. R. A. Nijboer, Physica 13, 605 (1947).
[CrossRef]

Pin Kao, H.

H. Pin Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

Rossi, M.

C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
[CrossRef]

Scarano, F.

F. Scarano, Meas. Sci. Technol. 24, 012001 (2013).
[CrossRef]

Segura, R.

C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
[CrossRef]

C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
[CrossRef]

Towers, C. E.

Towers, D. P.

van Kampen, N. G.

N. G. van Kampen, Physica 16, 817 (1950).
[CrossRef]

N. G. van Kampen, Physica 14, 575 (1949).
[CrossRef]

Verkman, A. S.

H. Pin Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 397/460/478.

Biophys. J.

H. Pin Kao and A. S. Verkman, Biophys. J. 67, 1291 (1994).
[CrossRef]

J. Visualization

C. Cierpka and C. J. Kähler, J. Visualization 15, 1 (2012).
[CrossRef]

Meas. Sci. Technol.

C. Cierpka, M. Rossi, R. Segura, and C. J. Kähler, Meas. Sci. Technol. 22, 015401 (2011).
[CrossRef]

C. Cierpka, R. Segura, R. Hain, and C. J. Kähler, Meas. Sci. Technol. 21, 045401 (2010).
[CrossRef]

F. Scarano, Meas. Sci. Technol. 24, 012001 (2013).
[CrossRef]

Opt. Lett.

Physica

N. G. van Kampen, Physica 14, 575 (1949).
[CrossRef]

B. R. A. Nijboer, Physica 13, 605 (1947).
[CrossRef]

K. Nienhuis and B. R. A. Nijboer, Physica 14, 590 (1949).
[CrossRef]

N. G. van Kampen, Physica 16, 817 (1950).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), pp. 397/460/478.

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Figures (5)

Fig. 1.
Fig. 1.

Optical setup for APTV. A cylindrical lens acting in the yz plane leads to the appearance of a second, separated focal plane. Particle image axis lengths, ax and ay, vary with distance to the camera (figure adopted from [3]).

Fig. 2.
Fig. 2.

Intensity coded pinhole/particle images in the central plane (intensity inverted; note that the intensity scaling differs between the images). Particle/pinhole images show an asteroid shaped diffraction pattern (high SNR provided).

Fig. 3.
Fig. 3.

Intensity coded pinhole/particle images in the focal plane and well beyond (intensity inverted; note that the intensity scaling differs between the images). Near the focal planes, the particle images form thin lines. Beyond the focal planes, the axis lengths increase in both directions.

Fig. 4.
Fig. 4.

Calibration function for a specific XY position. Both the minimum and maximum values of the axis ratio, ax/ay, denote focal planes (ΔzFP25mm).

Fig. 5.
Fig. 5.

Position error, Ez, depending on z location. (a) Average error, E¯z, is 0.133 mm (0.33%) at a measurement depth of 40 mm. (b) Using adjusted processing parameters, E¯z decreases to 0.075 mm (0.32%), when only the region between focal planes is considered (gray backgrounded area).

Equations (8)

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

U(P)=iλAeikRReik[Φ+s]sdS,
Φast=βastr2cos(2φ),
Φsph=βsphr4,
U(P)=C·02π0aexp[i(r2p+rqcos(φψ)βastr2cos(2φ)βsphr4)]rdrdφ,
C=ikA2πReikR.
I(P)=|U(P)|2.
ax=2a|p+βast|
ay=2a|pβast|.

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