## Abstract

A time-averaged recording of a sinusoidally vibrating object reconstructs a fringe function, which is determined by the ratio between the exposure time and the vibration period. For short exposures, compared with the vibration period, the fringe function is highly dependent on the number of vibration cycles recorded and on the starting point of the exposure in the vibration cycle. When several fringe functions that are recorded at different parts of the vibration cycle are added, the resulting averaged fringe function is similar to the normal *J*_{0}^{2} function, even at short exposures. The frequency ranges at which numerical analysis can be used in these two cases are defined, and the result of short exposures permitting digital fringe analysis, even under extremely unstable situations, is demonstrated. Extending the standard video exposure time permits recording vibrations at low frequencies as the normal *J*_{0}^{2} function and improves the light economy.

© 1993 Optical Society of America

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### Equations (4)

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(1)
$${U}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y\phantom{\rule{0.1em}{0ex}},t)={U}_{0}\phantom{\rule{0.1em}{0ex}}(x,y)exp\phantom{\rule{-0.1em}{0ex}}\left\{\phantom{\rule{0.1em}{0ex}}\left[\frac{4\pi i{a}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)}{\lambda}\right]sin\phantom{\rule{-0.1em}{0ex}}[2\pi {f}_{0}t+{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}]\phantom{\rule{0.1em}{0ex}}\right\}\phantom{\rule{0.1em}{0ex}},$$
(2)
$$I\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}{I}_{0}\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.2em}{0ex}}{\left|\phantom{\rule{0.1em}{0ex}}\frac{1}{(T-{T}_{2})\phantom{\rule{0.1em}{0ex}}}{\mathit{\int}}_{{T}_{1}}^{{T}_{2}}\phantom{\rule{0.1em}{0ex}}exp\phantom{\rule{-0.1em}{0ex}}\{\phantom{\rule{0.1em}{0ex}}\frac{4\pi i}{\lambda}{a}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}\times sin[2\pi {f}_{0}t\phantom{\rule{0.1em}{0ex}}+{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x,y)\phantom{\rule{0.1em}{0ex}}]\phantom{\rule{0.1em}{0ex}}\text{d}t\phantom{\rule{0.1em}{0ex}}\phantom{\rule{0.1em}{0ex}}\right|}^{2}\phantom{\rule{0.1em}{0ex}},$$
(3)
$$\begin{array}{ll}I\phantom{\rule{0.1em}{0ex}}(x,y)\hfill & ={I}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.2em}{0ex}}{\left|\phantom{\rule{0.1em}{0ex}}\text{\u2211}_{q\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}-\infty}^{\infty}\phantom{\rule{0.1em}{0ex}}{J}_{q}\phantom{\rule{0.1em}{0ex}}[u\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}]\phantom{\rule{0.1em}{0ex}}\times exp\phantom{\rule{-0.1em}{0ex}}\{iq\phantom{\rule{0.2em}{0ex}}[\phantom{\rule{0.1em}{0ex}}\overline{\theta}\phantom{\rule{0.1em}{0ex}}+{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}]\phantom{\rule{0.1em}{0ex}}\}sinc\phantom{\rule{0.1em}{0ex}}\left(q\phantom{\rule{0.1em}{0ex}}\frac{{T}_{c}}{{T}_{0}}\right)\phantom{\rule{0.1em}{0ex}}\right|}^{2}\hfill \\ \hfill & ={I}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}Z\phantom{\rule{0.1em}{0ex}}\{u\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}},{\phi}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}+\overline{\theta},{T}_{c}\}\phantom{\rule{0.1em}{0ex}},\hfill \end{array}$$
(4)
$$\begin{array}{ll}I\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)& ={I}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\text{\u2211}_{q\phantom{\rule{0.1em}{0ex}}=\phantom{\rule{0.1em}{0ex}}-\infty}^{\infty}{{J}_{q}}^{2}\phantom{\rule{0.1em}{0ex}}[u\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)]\phantom{\rule{0.1em}{0ex}}{sinc}^{2}\phantom{\rule{0.1em}{0ex}}\left(q\phantom{\rule{0.1em}{0ex}}\frac{{T}_{c}}{{T}_{0}}\right)\\ & ={I}_{0}\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}}{Z}_{\text{av}}\phantom{\rule{0.1em}{0ex}}\{u\phantom{\rule{0.1em}{0ex}}(x\phantom{\rule{0.1em}{0ex}},y)\phantom{\rule{0.1em}{0ex}},{T}_{c}\}\phantom{\rule{0.1em}{0ex}},\end{array}$$