Three-dimensional microscopic light field particle image velocimetry

Abstract

A microscopic particle image velocimetry (\(\mu \text {PIV}\)) technique is developed based on light field microscopy and is applied to flow through a microchannel containing a backward-facing step. The only hardware difference from a conventional \(\mu\)PIV setup is the placement of a microlens array at the intermediate image plane of the microscope. The method combines this optical hardware alteration with post-capture computation to enable 3D reconstruction of particle fields. From these particle fields, we measure three-component velocity fields, but find that accurate velocity measurements are limited to the two in-plane components at discrete depths through the volume (i.e., 2C-3D). Results are compared with a computational fluid dynamics simulation.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. 1126862. JB gratefully acknowledges funding from the Office of Naval Research under task number N0001413WX20545 monitored by program officer Dr. Ronald Joslin (ONR Code 333).

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Correspondence to Tadd T. Truscott.

Appendix

Appendix

We now derive the relationship between the synthetic image plane displacement, d and the corresponding object plane displacement, \(\delta,\) as given by Eq. 2 and shown schematically in Fig. 1a. The object plane is located at a distance

$$\begin{aligned} s_o^o = \delta +f_o \end{aligned}$$
(11)

to the left of the objective. The location of the image formed by the objective is found using the thin lens equation to give

$$\begin{aligned} s_i^o=\frac{f_o s_o^o}{s_o^o - f_o}, \end{aligned}$$
(12)

where \(s_i^o\) is positive to the right of the objective as drawn in Fig. 1a. The image formed by the objective becomes the object for the tube lens and is at a distance

$$\begin{aligned} s_o^t =\, & {} f_t+f_o-s_i^o\nonumber \\ =\, & {} f_t+f_o-\frac{f_o s_o^o}{s_o^o-f_o} \end{aligned}$$
(13)

from the tube lens (\(s_o^t\) is positive to the left of the tube lens). The thin lens equation is then applied to the tube lens to give the distance to the image plane, which coincides with the displaced \(s't'\) plane,

$$\begin{aligned} s_i^t = \frac{f_t\left( \frac{-f_o^2}{s_o^o-f_o}\right) +f_t^2}{\frac{-f_o^2}{s_o^o-f_o}} \end{aligned}$$
(14)

Inserting the relation for object plane displacement given in Eq. 11, Eq. 14 can be solved to give

$$\begin{aligned} s_i^t= & {} f_t-\left( \frac{f_t^2}{f_o^2}\right) \delta \nonumber \\= & {} f_t-M^2\delta, \end{aligned}$$
(15)

where the definition for magnification has been inserted. Finally, inserting the relation \(s_i^t = f_t-d\) into Eq. 15 and rearranging yields,

$$\begin{aligned} \delta = \frac{d}{M^2} \end{aligned}$$
(16)

As described in Sect. 2.1, the object plane displacements must be corrected to account for the fact that the index of refraction of the object medium differs from the medium in which the objective is immersed. Pereira and Gharib (2002) derived the correction for the apparent object depth Z, measured from the objective lens plane to the location at which the object would exist if there were no index of refraction changes. The actual object depth is given by

$$\begin{aligned} Z^\prime = D + w + \Omega \left( n_f \right) \left[ Z - D - \frac{w}{\Omega \left( n_w \right) } \right], \end{aligned}$$
(17)

where D is the distance from the objective plane to the channel wall, w is the thickness of the channel wall, \(n_w\) is the index of refraction of the channel wall and the function \(\Omega \left( \nu \right)\) is defined as

$$\begin{aligned} \Omega \left( \nu \right) = \sqrt{\frac{R^2}{Z^2} \left[ \left( \frac{\nu }{n}\right) ^2 - 1 \right] + \left( \frac{\nu }{n}\right) ^2} \end{aligned}$$
(18)

where \(R^2 = X^2 + Y^2\) is the radial coordinate of the object point with respect to the main optical axis and n is the index of refraction of the medium in which the objective is immersed (\(n = 1\) herein). For rather extreme values in our setup of \(R = 400\,\upmu\)m and \(Z = 1000\,\upmu\)m, we have \(\Omega (n_f ) = 1.353,\) which is approximately equal to \(n_f/n = 1.333.\) Therefore, we make an approximation that is often made (Tien et al. 2008; Galbraith 1955) and assume that \(\Omega (n_f ) \approx n_f/n,\) which conveniently removes the dependence of the depth correction on the radial coordinate. Thus, Eq. 17 reduces to

$$\begin{aligned} Z^\prime = \frac{n_f}{n} Z + \left( 1-\frac{n_f}{n} \right) D + \left( 1-\frac{n_f}{n_w} \right) w \end{aligned}$$
(19)

where we have also assumed that \(\Omega \left( n_w\right) \approx n_w/n.\) Therefore, the actual corrected depth distance between any two planes is given as:

$$\begin{aligned} \Delta Z^\prime = Z^\prime _2 - Z^\prime _1 = \frac{n_f}{n} \left( Z_2-Z_1\right) = \frac{n_f}{n} \Delta Z \end{aligned}$$
(20)

Substituting for \(\Delta Z\) the apparent object plane displacement \(\delta,\) we get

$$\begin{aligned} \delta ^\prime = \frac{n_f}{n} \delta \end{aligned}$$
(21)

which is Eq. 8.

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Truscott, T.T., Belden, J., Ni, R. et al. Three-dimensional microscopic light field particle image velocimetry. Exp Fluids 58, 16 (2017). https://doi.org/10.1007/s00348-016-2297-3

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Keywords

  • Particle Image Velocimetry
  • Point Spread Function
  • Light Field
  • Particle Tracking Velocimetry
  • Microlens Array