## Abstract

We describe a phase-shifting out-of-plane speckle interferometer
operating at 1 kHz for studying dynamic events. The system is based
on a Pockels cell that is synchronized to a high-speed video camera to
ensure that the phase shifting occurs between frames. Phase
extraction is performed by use of a standard four-frame algorithm, and
temporal phase unwrapping allows sequences of several hundred absolute
(rather than relative) displacement maps to be obtained fully
automatically. The maximum theoretical surface velocity of 67
µm s^{-1} is a factor of 40 greater than can be
achieved with a speckle interferometer based on a conventional video
camera. We test the system using a target that is displaced with
constant speed in a direction normal to its surface by means of a
piezoelectric transducer. The system’s performance in a practical
situation is illustrated with measurements on a thin plate undergoing
out-of-plane deformation.

© 1999 Optical Society of America

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### Equations (18)

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(1)
$$I\left(t\right)={I}_{r}+{I}_{o}+2\sqrt{{I}_{r}{I}_{o}}cos\left[\mathrm{\Phi}\left(t\right)+\mathrm{\varphi}\left(t\right)\right],$$
(2)
$$\mathrm{\varphi}\left(t\right)=\mathrm{\pi}t/2.$$
(3)
$${\mathrm{\Phi}}_{w}\left(t\right)={tan}^{-1}\left[\frac{N\left(t\right)}{D\left(t\right)}\right],$$
(4)
$$N\left(t\right)=\mathrm{Im}\left[z\left(t\right)\right],D\left(t\right)=\mathrm{Re}\left[z\left(t\right)\right],$$
(5)
$$z\left(t\right)=\left\{\left[I\left(t\right)-I\left(t+2\right)\right]+i\left[I\left(t+3\right)-I\left(t+1\right)\right]\right\}\times exp\left(-i\mathrm{\pi}t/2\right).$$
(6)
$$\mathrm{\Phi}\left(t+h\right)\approx \mathrm{\Phi}\left(t\right)+h\mathrm{\Phi}\prime \left(t\right),h=1,2,\dots q-1,$$
(7)
$$\mathrm{\varphi}\left(t+h\right)=\mathrm{\varphi}\left(t\right)+h\left(\mathrm{\pi}/2\right).$$
(8)
$$\mathrm{d}\left(t\right)=\mathrm{NINT}\left\{\left[{\mathrm{\Phi}}_{w}\left(t\right)-{\mathrm{\Phi}}_{w}\left(t-1\right)\right]/2\mathrm{\pi}\right\},t=1,2,\dots s,$$
(9)
$$\mathit{\nu}\left(t\right)=\sum _{t\prime =1}^{t}\mathrm{d}\left(t\prime \right),t=1,2,\dots s,$$
(10)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left(t,0\right)={\mathrm{\Phi}}_{w}\left(t\right)-{\mathrm{\Phi}}_{w}\left(0\right)-2\mathrm{\pi}\mathit{\nu}\left(t\right).$$
(11)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{w}\left({t}_{2},{t}_{1}\right)={tan}^{-1}\left[\frac{N\left({t}_{2}\right)D\left({t}_{1}\right)-D\left({t}_{2}\right)N\left({t}_{1}\right)}{D\left({t}_{2}\right)D\left({t}_{1}\right)+N\left({t}_{2}\right)N\left({t}_{1}\right)}\right].$$
(12)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left(t,0\right)=\sum _{t\prime =1}^{t}\mathrm{\Delta}{\mathrm{\Phi}}_{w}\left(t\prime ,t\prime -1\right).$$
(13)
$$A\left(x,y\right)=\frac{1}{9}\left[\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right].$$
(14)
$$\mathrm{\sigma}=1.2\mathrm{\lambda}F\left(1+M\right)$$
(15)
$$\mathrm{d}\left(t\right)=\mathrm{NINT}\left\{\left[\mathrm{\Delta}{\mathrm{\Phi}}_{w}\left(t,0\right)-\mathrm{\Delta}{\mathrm{\Phi}}_{w}\left(t-1,0\right)\right]/2\mathrm{\pi}\right\},t=2,3,\dots s.$$
(16)
$$\mathit{\nu}\left(t\right)=\sum _{t\prime =2}^{t}\mathrm{d}\left(t\prime \right),t=2,3,\dots s,\mathit{\nu}\left(1\right)=0,$$
(17)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left(t,0\right)=\mathrm{\Delta}{\mathrm{\Phi}}_{w}\left(t,0\right)-2\mathrm{\pi}\mathit{\nu}\left(t\right),t=1,2,\dots s.$$
(18)
$$\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left(t,0\right)=\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left(t,{t}_{\mathrm{\kappa}}\right)+\sum _{k=2}^{\mathrm{\kappa}}\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left({t}_{k},{t}_{k-1}\right)+\mathrm{\Delta}{\mathrm{\Phi}}_{u}\left({t}_{1},0\right).$$