Particle image velocimetry: rapid transparency analysis using optical correlation Jeremy M. Coupland and Neil A. Halliwell

University of Southampton, Institute of Sound & Vibra­tion Resarch, Southampton S09 5NH, U.K. Received 22 March 1988. 0003-6935/88/101919-03$02.00/0. © 1988 Optical Society of America.

Particle image velocimetry (PIV)1 provides a means of measuring instantaneous 2-D velocity vector fields from a plane of interest within an unsteady flow. The technique was pioneered by Dudderar and Simpkins2 in 1977 and has since been used to study convection currents,3 flow around unexcited jets,4 and more recently breaking wave phenon-ema.5

In its simplest form, PIV uses double-exposure photogra­phy to record the position of seeding particles contained within a thin sheet of light illuminating a section of the flow. The resulting transparency is divided into a grid of small interrogation regions which are systematically analyzed to obtain the average displacement of the particle images re­corded within each region. If the time interval between

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exposures and the magnification of the recording optics are known, these displacement measurements can be related to the fluid velocity vectors and a 2-D flow map constructed.

Since there are typically several thousand interrogation regions on each PIV transparency, it is essential that any analysis procedure be fast, accurate, and easily automated. Fringe analysis6 has been used successfully to take similar measurements from double-exposure speckle photographs and has been found to work well in the PIV situation provid­ing a sufficiently large number of particle image pairs N are within the interrogation region (N > 6).7 As N decreases the fringe visibility is reduced and direct measurement of the fringe orientation and spacing becomes more difficult. In these cases many workers compute the 2-D Fourier trans­form of the fringe pattern to find the main frequency compo­nent.8 This operation is equivalent to computing the spatial autocorrelation of the intensity distribution immediately behind the illuminated interrogation region. Two-dimen­sional Fourier transformation is computationally intensive and is not, therefore, a realistic option for an automated system. A second method of data reduction is to compress the 2-D image onto orthogonal (X, Y) axes. One-dimension­al spatial correlation can then extract the X and Y compo­nents of the average particle displacement. This method has been found to work well in practice providing N is small (N < 3).9

In summary, PIV works well providing the seeding density is sufficient for fringe analysis or low enough for 1-D com­pression techniques to be used. In practical situations, ran­dom fluctuations in seeding density frequently mean that neither approach is applicable. Correlation analysis can be used regardless of the seeding density but at the expense of computation time.

In this Letter we report an optical method for computing 2-D autocorrelation directly, thus providing a fast and accu­rate analysis procedure that is capable of working well over a wide range of seeding densities.

The design of the correlator originates from the work of VanderLugt who demonstrated the synthesis of an optical matched filter using holographic methods.10 Although this work was published in 1963 it has only met very specialist applications since processing and accurate repositioning of the filter are necessary. The recent development of photore­fractive materials has renewed interest in optical correlation, since holograms can now be recorded and erased in situ, and several designs of photorefractive correlator have been pub­lished.11,12 In this Letter we report a new design, which is simple to construct and well suited to PIV analysis.

A schematic diagram of the optical correlator is shown in Fig. 1. A crystal of bismuth silicon oxide (BSO) was chosen

Fig. 1. Experimental arrangement.

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for this work, since it is readily available and displays a reasonably large photorefractive coefficient when an exter­nal electric field is applied. The correlation is split into two distinct operations. First, the VanderLugt filter is recorded as a Fourier plane hologram of the light amplitude immedi­ately behind the illuminated interrogation region. The ref­erence beam is subsequently switched off, and the autocorre­lation function of this region is formed in the output plane.

The holographic grating formed within the optically thick BSO crystal is described in full by the coupled-mode equa­tions of Kogelnik.13 However, provided the Bragg condition is satisfied, this theory is reduced to that of thin holographic gratings, which is presented here. For a discussion of the Bragg condition with reference to optical correlation the reader is referred to the work of Pepper et al.14

With reference to Fig. 1, according to the Fourier trans­forming properties of convex lenses,15 the complex ampli­tude U2(x2,y2) of the field in the crystal plane can be written as the sum of two components:

where A(k1x2,k1y2) is the Fourier transform of the input field a(x1,y2), and B(x2,y2) is the reference field. The constant k1 is given by k1 = 1/λƒ1, where λ is the wavelength of the laser illumination. This field distribution gives rise to a refrac­tive-index change in the crystal proportional to the intensity:

The resulting filter is described by the complex transmission function T(x2,y2):

where B is a constant describing the photorefractive efficien­cy of the crystal. Since the photoinduced index modulation is small, i.e., βI « 1, the transmission function can be written

When the reference beam is switched off the optical field immediately after the filter is given by

The last term is of interest since it describes a reconstruction of the reference beam modulated by the squared modulus of the Fourier transformed input field. Accordingly, the out­put field of interest U3(x3,y3) is this term transformed by lens 2 to give

where * and ** denote convolution and correlation, respec­tively, and the constant k2 is given by k2 = 1/λƒ2. If the reference beam is a plane wave, b(x3,y3) is a delta function on which the autocorrelation of a(x3,y3) is centered. Choosing the position of this delta function as the origin of the coordi­nate system, we can write the intensity distribution in the output plane, I3(x3,y3), as

Fig. 2. (a) Input function; (b) output autocorrelation function; (c) computed autocorrelation function.

This distribution is captured by a TV camera, and the posi­tion of the largest noncentral peak can easily be measured to find the particle velocity vector.

An example of an optical correlation is shown in Fig. 2. The input function, the optical autocorrelation, and an iso­metric plot of the output intensity computed digitally are included for comparison. These photographs were taken using a polished BSO crystal measuring 9 X 10 X 2 mm with electrodes applied to the 10- X 2-mm edges to produce an electric field of 6 kV in the 110 direction. Since the photore­fractive effect saturates when the crystal has absorbed suffi­cient energy, there is a trade-off between illumination inten­sity and response time. In this case the illumination power was of the order of 10 mW (λ = 514.5 nm) giving a response time of ~ 1 s.

An interesting extension of PIV analysis technique using optical correlation arises due to increased processing speed. By interrogating overlapping areas of the PIV transparency, the position of the correlation peak can be tracked as the transparency is scanned, thus avoiding ambiguity prob­lems.16

A design of an optical correlator suitable for rapid PIV data reduction has been demonstrated. The inherent speed advantage of this system over comparable digital techniques facilitates more extensive analysis of each transparency re­ducing ambiguity related error. Further work to define the accuracy and limitations of the technique are published in a subsequent full paper.

References 1. C. J. D. Pickering and N. A. Halliwell, "Particle Image Veloci-

metry: A New Field Measurement Technique," in Optical Measurement in Fluid Mechanics, Inst. Phys. Conf. Series 1977 (Adam Hilger, Bristol, 1985).

2. T. D. Dudderar and P. G. Simpkins, "Laser Speckle Photogra­phy in a Fluid Medium," Nature London 270,45 (1977).

3. T. D. Dudderar and P. G. Simpkins, "Laser Speckle Measure­ments of Transient Bernard Convection," J. Fluid Mech. 89, 665 (1978).

4. R. Meynart, "Instantaneous Velocity Measurement in Unsteady Gas Flow by Speckle Photography," Appl. Opt. 22, 535 (1983).

5. I. Grant, "Flow Measurement by Pulsed Laser (Speckle)," Phys. Bull. 38, 175 (1987).

6. C. J. D. Pickering and N. A. Halliwell, "PIV: Fringe Visibility and Pedestal Removal," Appl. Opt. 24, 2474 (1985).

7. R. J. Adrian, "Multi-point Optical Measurement of Simulta­neous Vectors in Unsteady Flow—a Review," Int. J. Heat Fluid Flow 7, 127 (1986).

8. J. M. Huntley, "An Image Processing System for the Analysis of Speckle Photographs," J. Phys. E 19, 43 (1986).

9. R. J. Adrian and C-S. Yao, "Orthogonal Compression and 1-D Analysis Technique for Measurement of Particle Displacements in Pulsed Laser Velocimetry," Appl. Opt. 23, 1687 (1984).

10. A. VanderLugt, "Signal Detection by Complex Spatial Filter­ing," IEEE Trans. Inf. Theory IT-10, 139 (1964).

11. J. O. White and A. Yariv, "Real-Time Image Processing via Four-wave Mixing in a Photorefractive Medium," Appl. Phys. Lett. 37, 5 (1980).

12. L. Pichon and J. P. Huignard, "Dynamic Joint-Fourier-Trans­form Correlator by Bragg Diffraction in Photorefractive Biι2SiO20 Crystals," Opt. Commun. 36, 227 (1981).

13. H. Kogelnik, "Coupled Wave Theory for Thick Hologram Gra­tings," Bell Syst. Tech. J. 48, 2909 (1969).

14. D. M. Pepper, J. AuYeung, D. Fekete, and A. Yariv, "Spatial Convolution and Correlation of Optical Fields via Degenerate Four-Wave Mixing," Opt. Lett. 3, 7 (1978).

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

16. J. M. Coupland, C. J. D. Pickering, and N. A. Halliwell, "Particle Image Velocimetry: Theory of Directional Ambiguity Removal Using Holographic Image Separation," Appl. Opt. 26, 1596 (1987).

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