Quantitative Speckle Photography

In the discussions that follow, it is assumed that a double exposure speckle photograph has been taken of a relatively flat, diffusively reflecting surface that was compressed vertically between exposures. The specklegram was obtained by imaging the surface onto a film plate located in front of the surface using a lens of suitable focal length. The illumination used was an expanded beam from a HeNe laser. The setup for obtaining the specklegram is shown at the right. Analysis of the specklegram on the film plate can be performed by either of two methods. The Young's fringes method is a point-by-point technique that gives the magnitude of in-plane motion and the line-of-motion, but not the signed direction. This sign must be determined by additional knowledge external to the specklegram. The other method utilizes Fourier filtering and is a full-field method that gives information about a particular size of displacement component. For instance, if the surface was basically compressed, proper placement of a filter aperture in the Fourier plane will produce a set of fringes that show, for example, the loci of all points on the surface whose total in-plane motion had a component of 8 microns along a line oriented at 45 degrees to the left of vertical.

Young's Fringes

This is a simple-to-implement point-by-point method of analysis that gives information about the direction and magnitude of the speckle displacement between the two exposures of the specklegram. The sign of the motion is unknown and must be determined from information obtained extrinsic to the specklegram. In the sketch at the right, assume that a small pencil of light of wavelength l is transmitted through a selected region of a double-exposure specklegram. A screen is located a distance D from the specklegram. Because of the 'twinning' effect of the before-and-after double-exposure, a set of Young's fringes is seen on the screen. By measuring the fringe spacing d and observing the orientation of the in-plane normal to the fringe lines, it is possible to determine the speckle spacing p. The calculation involved starts with the Young interference equation: p sin q = l. For the small angles usually involved, sin q = d/D, so the speckle displacement p = lD/d. This is not yet the displacement of points on the original surface because of the magnification involved in imaging the original surface onto the film plate using the lens. To get the magnification factor, recall that 1/f = 1/s + 1/s' and m = -s'/s, where s and s' are the object and image distances to the lens, f is the lens focal length, and m is the (image/object) height ratio (with a minus sign if the image is inverted). Scaling the speckle displacement p by this magnification factor m will give the displacement on the object surface of points at the location imaged by the region of the specklegram which is illuminated by the small-diameter laser beam. The direction of the normal to the speckle fringe pattern shows the line-of-motion along which this displacement occurred, but the actual direction of motion must be determined from knowledge other than the specklegram.

Fourier Filtering

This method of analysis gives full-field information about speckle displacements, but only for particular displacement components in a chosen direction. The setup for the procedure is shown in the figure below:

The key to this procedure is the positioning of the aperture in the Fourier filter. The sensitivity of the measurement depends on the distance of the aperture from the center axis. For instance, if a small aperture A is located at a distance r from the optic axis of the image, then a speckle spacing of p and an orientation of q with respect to the x axis anywhere in the specklegram grating will cause light to be sent through the aperture A in the Fourier filter located at the focal plane of the lens if p = nlf/r, where n is some integer, and f is the focal length of the lens. Note that a different speckle spacing p' and different orientation q' will send light through the aperture if the displacement component of that speckle pair measured along the direction of the first speckle pair satisfies p = p' sin f, where f = |q' - q |. Thus the observed fringes are the loci of constant displacement components which satisfy the given requirement. Again, the specklegram does not give the actual direction of displacement, but only the line-of-motion. The actual direction must be determined from external knowledge and cannot be obtained from the specklegram. Note that by moving the aperture in and out along a radial line, the sensitivity of the measurement can be changed. The point is that the sensitivity is changed not during data acquisition, but during data analysis. Finally, note again that the measurement of speckle displacement p must be multiplied by the magnification m due to the lensing setup.

The Fourier filter method seems to give more information over the entire surface than the Young's fringes method, but it is more difficult in practice than the other method because of the difficulty of accurately positioning the filter aperture. Also, if the aperture is kept small, the amount of light that passes through it is also small, and a rather lengthy time exposure is required to obtain the desired fringe pattern.

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