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.
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.