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We are developing an analog IC that implements the key computation of double-exposure heterodyne laser interferometry, a technique recently developed by Voelz et al. and McMackin et al. that measures the profile of a reflective surface using wavefront sensing.

First a laser beam is split. The reference beam, Eref(t), is directed onto an imaging array via a fixed-length path. The probe beam, E(r,t), is passed through an acousto-optic modulator that shifts its frequency by a small, controlled amount, and then it is directed so as to illuminate an object. The light reflected from the object is focused onto the same imaging array as the reference beam, resulting in an interference pattern that modulates at the difference between the two beams (i.e., the beat frequency). The phase of this beating is related to the line-of-sight distance traveled by the probe beam and thus the line-of-sight distance from the imager to points on the illuminated surface. The phase may be recovered from measurements over time of the varying light intensity at each pixel.

Following Voelz et al., the intensity at the sensor is given by

where A(r,t) and Aref(t) are the amplitudes of the separate beams, Dw is the beat frequency, and fref + f(r) is the phase. By sampling this image N times over a period tn, we can then find the real and imaginary components of the Fourier coefficient of the beat frequency at the mth pixel location, rm, with

 

The phase map can then be computed from

If the phase measurement is synchronized with the modulation that produces the different illuminant frequencies, then two phase maps taken at different times can be compared to obtain a measure of the motion or deformation of the illuminated object between the two measurements, or "exposures." Specifically, the difference in the two phase values at a pixel gives a measure of line-of-sight displacement of the corresponding point in the image, modulo the wavelength of the illuminating light. Moreover, spatial (pixel-to-pixel) differences in surface displacement of a deformed object are a local measure of deformation.

Because the light used in this technique is in the visible spectrum, the displacements that can be measured without spatial aliasing are limited to very small optical wavelengths (514 nm was used in Voelz et al.). To circumvent this constraint, a double-exposure technique is presented in McMackin et al., which is analogous to that just described, except that the frequency of the source laser is slightly shifted between exposures. The net result of taking the phase difference in this case is a measure of line-of-sight distance from imager to object, modulo a much longer "equivalent wavelength" than the physical wavelengths of the illuminants. In an obvious extension of the displacement measurement technique, two sets of such double exposures with frequency-shifting, one before and the other after an object moves, could be used to measure displacements or deformations when they are expected to be greater than the illuminant wavelength.

 

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