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  • Editor: Xi-Cheng Zhang
  • Vol. 39, Iss. 5 — Mar. 1, 2014
  • pp: 1298–1301
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Three-dimensional location of micrometer-sized particles in macroscopic domains using astigmatic aberrations

Thomas Fuchs, Rainer Hain, and Christian J. Kähler  »View Author Affiliations


Optics Letters, Vol. 39, Issue 5, pp. 1298-1301 (2014)
http://dx.doi.org/10.1364/OL.39.001298


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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

In the last few years, spatial particle location using astigmatic aberrations was applied increasingly to measure three-dimensional flow fields in microfluidics [1

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

]. The astigmatism paradigm was first introduced in 1994 to track single fluorescent particles in artificial solutions and living cells [2

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

]. Astigmatism leads to the appearance of two separated focal planes. This distance depends mainly on the amount of astigmatism induced by a cylindrical lens placed in front of the camera sensor. Because of this aberration, a tracer particle forms an elliptical particle image with a distinct geometry corresponding to its distance from the camera (Fig. 1). Hence, the z position of a particle is coded in the horizontal and vertical axis length (ax and ay) of its elliptical particle image [4

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

].

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]).

There is considerable interest to employ astigmatism particle tracking velocimetry (APTV) and its unique capabilities for the measurement of macroscopic flows. Unlike multicamera techniques such as tomographic PIV [5

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

], APTV employs only a single camera. Thus, APTV is best suited for measurement domains with limited optical access (e.g. compressor, turbine, combustion chamber, and engine research). In a study presented by Towers and Towers [6

6. C. E. Towers and D. P. Towers, Opt. Lett. 31, 1220 (2006). [CrossRef]

] astigmatic aberrations are used in a macroscopic particle position determination approach. Geometrical optics approaches are used to deduce depth information from particle image dimensions. Aberrations besides astigmatism are not accounted for in this geometrical model, leading to errors in depth position determination. Sensitivity to illumination disparities introduces further inaccuracies. The measurement depth range is limited to regions well within the two focal planes as the geometrical model does not apply in near-focal regions. Therefore, the distance of the focal planes has to be increased to measure a certain depth range, resulting in larger particle images with lower SNRs.

This Letter presents a theoretical and experimental image formation study in the presence of astigmatic aberrations. Particles are simulated by pinholes as they have a similar light emission behavior. In the second part, a three-dimensional, macroscopic location scheme of micrometer-sized particles is introduced for the APTV technique. The technique’s feasibility to measure macroscopic flows is proven by an experimental accuracy analysis of the particle z position determination, using state-of-the-art PIV equipment.

The image formation study provides a qualitative description of particle image shapes in the presence of astigmatism. In macroscopic APTV setups astigmatic aberrations, induced by a cylindrical lens placed in front of the camera sensor, can be considered as large aberrations, i.e., the deviation of the actual wavefront from the ideal, spherical wavefront is large. Thus, geometrical optics describe image formation relatively accurately. Nonetheless, for a comprehensive analysis of the image formation, the influence of diffraction on image formation has to be accounted for. According to the Huygens–Fresnel principle, the diffraction integral for a disturbance at a point P in image space, in the presence of aberrations, is given by [7

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

]
U(P)=iλAeikRReik[Φ+s]sdS,
(1)
where λ is the wavelength, k is the wavenumber, A is the amplitude at Q (point on Gaussian reference sphere), R is the radius of the Gaussian reference sphere, s is the distance between Q and P, S is the wavefront surface, and Φ denotes the aberration function. This aberration function describes the deviation of wavefronts from ideal, spherical wavefronts at the exit pupil. If Φ=0, i.e., no aberrations in the optical system are present, the solution of Eq. (1) is the Airy-pattern [7

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

]. In the case of astigmatism, the aberration function is given by
Φast=βastr2cos(2φ),
(2)
where βast denotes the quantitative deviation of the aberrated wavefront from the ideal one, and r and φ are polar-coordinates of Q. For spherical aberrations, the aberration function is
Φsph=βsphr4,
(3)
where the amount of spherical aberration is denoted by βsph. By introducing the coordinates p (along the optical axis), and the polar-coordinates q, and ψ for P in image space, the diffraction integral in the presence of astigmatic and spherical aberration is formulated as [8

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

]
U(P)=C·02π0aexp[i(r2p+rqcos(φψ)βastr2cos(2φ)βsphr4)]rdrdφ,
(4)
where a is the exit pupil radius, and C is described by
C=ikA2πReikR.
(5)
For small aberrations, i.e., the deviation of the actual wavefront from the ideal wavefront being less than λ, the diffraction image can be calculated by expanding the aberration function in a series of circular polynomials [9

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

]. An experimental verification of this treatment was given by Nienhuis in 1949 [10

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

]. If aberrations are somewhat larger, i.e., the deviation is several times λ, an asymptotic expansion of the diffraction integral can be applied to calculate the diffraction image [11

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

]. For the presented analysis, Matlab was used to solve Eq. (4) numerically. The resulting intensity at P is then calculated as
I(P)=|U(P)|2.
(6)

In the presence of astigmatism, the location of the plane of least distortion, i.e., midway between the focal planes, is at p=0. Particle images show asteroid shapes with a circular ring of higher intensity (Fig. 2, and [5

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

,7

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

,8

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

]). The image of the 5 μm pinhole in Fig. 2(b) matches very well with the simulated image in Fig. 2(a). The asteroid shape can hardly be observed with a lower SNR [image of the 1 μm pinhole, Fig. 2(c)]. The image has a circular shape of almost equi-distributed intensity with a less distinct outer ring. The geometrical theory of aberrations predicts a circular image of constant intensity in the central plane, quite similar to the image analyzed in Fig. 2(c). The distance of the two focal planes is determined by the amount of astigmatism. Their location is denoted by p=±βast along the optical axis. Images at the focal planes form thin lines of high intensity [Figs. 3(a) and 3(b)], whereas geometrical optics predicts one axis length to be zero (provided ideal lenses are present) [10

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

]:
ax=2a|p+βast|
(7)
ay=2a|pβast|.
(8)
Beyond the focal planes, i.e., p>βast or p<βast, both axes grow with a bright outer ring and less accentuated rings in the particle image center [Figs. 3(c) and 3(d)].

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. 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.

Spherical aberrations have only limited influence on image formation in macroscopic domains, as the amount of spherical aberrations is small compared to the induced astigmatic aberration (βastβsph). Therefore, spherical aberrations can be neglected for the image formation analysis in macroscopic domains. In microscopic domains, the influence of spherical aberrations on the image formation is significant, as the focal plane distances are small (βastβsph). In macroscopic domains, the appearance of particle images can be considered to be symmetric with respect to p=0 but rotated by 90 degrees. In microscopic domains, particle images beyond one focal plane (p>βast) have a bright outer ring and beyond the other focal plane (p<βast) they have a bright central spot with decreasing intensity toward the particle image edge. With a sufficiently small amount of astigmatism, the asteroid-shaped diffraction pattern cannot be observed anymore. Instead particle images have cross-shaped central spots in the center. In this case, the location of the focal planes cannot be determined as easily, as particle images do not form thin lines anymore.

To account for particle image shapes as observed in macroscopic APTV setups, a suitable image processing algorithm is established. Local intensity distributions at the four vertices of the elliptical particle images are fitted by means of a thin-plate spline. An intensity threshold denotes the subpixel locations of ellipse vertices, from which center locations and axis lengths, ax and ay, are determined. Local intensity distributions are normalized by particle images’ mean intensities. Hence, differing illumination intensities and distributions have a small impact on the position determination, making the processing much more robust. The image formation study has proven that in macroscopic APTV setups image formation is dominated by astigmatism, leading to similar-shaped particle images. As a consequence, the macroscopic APTV processing and calibration procedures are applicable independently of the measurement setup.

The geometry of a pinhole image does not only depend on the distance to the camera (z position) but also on its x and y position (X and Y position, respectively), as the focal planes are curved. In addition, distortions in the optical path have to be accounted for. Hence, it is necessary to establish a calibration function depending on the geometry of a pinhole image and the spatial coordinates of the corresponding pinhole. The first step of the calibration procedure is to calculate an intensity averaged image of the 20 recordings at each z position. In a second step, spline fits are applied to the axis ratio values of the pinhole images detected on the averaged recordings. From these spline fits, unique calibration functions for every X and Y position, as seen in Fig. 4, can be determined. According to this calibration procedure, z positions of particle/pinhole images are estimated as follows: center locations and axis ratios are determined first. A calibration function (X and Y position denoted by center location) for each single particle/pinhole image is derived from the spline fits of the averaged images. With this unique calibration function, z positions of particle/pinhole images are approximated by the corresponding axis ratios. When measuring beyond focal planes, the calibration functions become ambiguous. These ambiguities can be overcome by analyzing the axis lengths of particle images. Whereas axis ratios have a minimum or maximum at focal planes, the larger axis of a particle image changes linearly near the corresponding focal plane. Thus, the length of the larger particle image axis determines whether the particle position is located within or beyond the focal planes.

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).

For the z position accuracy analysis, the absolute deviations, Δzp, of the estimated positions compared to the actual positions are determined. At every z location the standard deviation of all Δzp values to the exact position (zero deviation) is calculated, denoted by the z position determination error Ez. For each z location, a data set of about 300 particle images is analyzed. Figure 5(a) shows the distribution of Ez along z, at a measurement depth of 40 mm. At both focal planes (approximate z positions: 8 and 32 mm), Ez has peaks even though SNRs are large. This can be explained by the calibration function that has a maximum or minimum at the respective focal planes (Fig. 4). Generally, Ez increases with decreasing slopes of the calibration functions. Lower SNRs yield higher Ez values as well. The average error, E¯z, is 0.133 mm (0.33%). An adjustment of the processing parameters improves the Ez values of the gray backgrounded area, i.e., only the area between the focal planes is considered. Large particle images with low SNRs located beyond the focal planes cannot be processed with these parameters. Distribution of Ez is then relatively uniform and follows the slope of the calibration function [Fig. 5(b)]. For a resulting measurement depth of 23.5 mm E¯z is 0.075 mm (0.32%).

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).

The relative z position determination accuracy of the proposed macroscopic APTV technique lies in the same range of the accuracy of the aforementioned anamorphic approach [5

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

]. For the latter approach no pulsed laser was used, allowing for longer exposures (higher SNRs) of the sensor. These long exposures, however, are not feasible for the measurement of macroscopic flows.

A theoretical and experimental image formation study was presented. Particle image processing algorithms were optimized on the basis of image formation observations. A three-dimensional, macroscopic location scheme of micrometer-sized particles, introduced in the second part of the Letter, showed small errors in particle z position determination. Average errors were as low as 0.33%, at a measurement depth of 40 mm. These accuracies prove the ability of APTV to measure volumetric velocity fields in macroscopic domains with only a single camera for applications with limited optical access.

The investigations were conducted as part of the joint research programme AG Turbo 2020 in the frame of AG Turbo. The work was supported by the Bundesministerium für Wirtschaft und Technologie (BMWi) as per resolution of the German Federal Parliament under grant number 03ET2013M. The authors gratefully acknowledge AG Turbo and MTU Aero Engines AG for their support and permission to publish this Letter. The responsibility for the content lies solely with its authors.

References

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]

OCIS Codes
(100.2960) Image processing : Image analysis
(100.6890) Image processing : Three-dimensional image processing
(110.2990) Imaging systems : Image formation theory
(260.1960) Physical optics : Diffraction theory

ToC Category:
Imaging Systems

History
Original Manuscript: October 24, 2013
Manuscript Accepted: January 4, 2014
Published: February 26, 2014

Citation
Thomas Fuchs, Rainer Hain, and Christian J. Kähler, "Three-dimensional location of micrometer-sized particles in macroscopic domains using astigmatic aberrations," Opt. Lett. 39, 1298-1301 (2014)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-39-5-1298


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References

  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]

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