## Abstract

A method for whole-field noncontact measurement of displacement, velocity, and acceleration of a vibrating object based on image-plane digital holography is presented. A series of digital holograms of a vibrating object are captured by use of a high-speed CCD camera. The result of the reconstruction is a three-dimensional complex-valued matrix with noise. We apply Fourier analysis and windowed Fourier analysis in both the spatial and the temporal domains to extract the displacement, the velocity, and the acceleration. The instantaneous displacement is obtained by temporal unwrapping of the filtered phase map, whereas the velocity and acceleration are evaluated by Fourier analysis and by windowed Fourier analysis along the time axis.
The combination of digital holography and temporal Fourier analyses allows for evaluation of the vibration, without a phase ambiguity problem, and smooth spatial distribution of instantaneous displacement, velocity, and acceleration of each instant are obtained. The comparison of Fourier analysis and windowed Fourier analysis in velocity and acceleration measurements is also presented.

© 2007 Optical Society of America

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### Equations (87)

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(5)
$${I}_{H}\left(x,y\right)={\left|{R}_{H}\left(x,y\right)\right|}^{2}+{\left|{U}_{H}\left(x,y\right)\right|}^{2}+{R}_{H}\left(x,y\right)\times {U}_{H}*\left(x,y\right)+{R}_{H}*\left(x,y\right){U}_{H}\left(x,y\right)\text{,}$$
(6)
{\theta}_{\mathrm{max}}
(7)
$${f}_{\mathrm{max}}=\frac{2}{\lambda}\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{{\theta}_{\mathrm{max}}}{2}\right)\text{,}$$
(8)
{f}_{\mathrm{max}}<1/\left(2\Delta \right)
(9)
{U}_{H}\left(x,y\right)
(10)
{U}_{H}\left(x,y;\text{\hspace{0.17em}}t\right)
(11)
\Delta \varphi ={\varphi}_{{t}_{2}}-{\varphi}_{{t}_{1}}
(12)
$$\Delta \varphi =\frac{2\pi z}{\lambda}\cdot S\text{,}$$
(18)
$$\Delta \varphi =\mathrm{arctan}\text{\hspace{0.17em}}\frac{\mathrm{Im}\left[U\left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)U*\left(x,y;\text{\hspace{0.17em}}{t}_{1}\right)\right]}{\mathrm{Re}\left[U\left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)U*\left(x,y;\text{\hspace{0.17em}}{t}_{1}\right)\right]}\text{,}$$
(20)
\left[0,2\pi \right)
(21)
U\left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)U*\left(x,y;\text{\hspace{0.17em}}{t}_{1}\right)
(22)
\phi \left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)
(24)
U\left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)U*\left(x,y;\text{\hspace{0.17em}}{t}_{n-1}\right)
(25)
\varphi \left(x,y;\text{\hspace{0.17em}}t\right)
(27)
\vartheta \left(x,y;\text{\hspace{0.17em}}t\right)=\mathrm{d}\varphi /\mathrm{d}t
(28)
\mathrm{exp}\left(j\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\vartheta \right)
(30)
\left(-\pi ,\pi \right)
(31)
\left(-\pi ,\pi \right)\text{\hspace{0.17em}}\text{rad}/\text{s}
(32)
$$Sf\left(u,\xi \right)={\displaystyle {\int}_{-\infty}^{+\infty}f\left(t\right){g}_{u\mathrm{\text{,}}\xi}*}\left(t\right)\mathrm{d}t\text{,}$$
(33)
$$f\left(t\right)=\frac{1}{2\pi}\text{\hspace{0.17em}}{\displaystyle {\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}Sf\left(u,\xi \right){g}_{u\text{,}\xi}}\left(t\right)\mathrm{d}\xi \mathrm{d}u}\text{,}$$
(34)
Sf\left(u,\xi \right)
(35)
{g}_{u\text{,}\xi}\left(t\right)
(36)
$${g}_{u\text{,}\xi}\left(t\right)=g\left(t-u\right)\mathrm{exp}\left(j\xi t\right)\text{.}$$
(37)
$$g\left(t\right)=\mathrm{exp}\left(-{t}^{2}/2{\sigma}^{2}\right)\text{,}$$
(38)
\Delta \varphi \left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)=\mathrm{arg}\left[U\left(x,y,{t}_{n}\right)U*\left(x,y,{t}_{1}\right)\right]
(39)
n=1,2\text{\hspace{0.17em}}\dots \text{\hspace{0.17em}}N
(41)
{C}_{P}=\mathrm{exp}\left(j\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}{\phi}_{P}\right)
(42)
$$S{C}_{p}\left(u,\xi \right)=\frac{\sqrt{s}}{2}\text{\hspace{0.17em}}{A}_{xy}\left(u\right)\mathrm{exp}\left\{j\left[\phi \left(u\right)-\xi u\right]\right\}\times \left(\widehat{g}\left\{s\left[\xi -\phi \prime \left(u\right)\right]\right\}+\epsilon \left(u,\xi \right)\right)\text{,}$$
(43)
{g}_{s}\left(t\right)={s}^{-1/2}g\left(t/s\right)
(44)
{A}_{xy}\left(u\right)
(46)
{A}_{xy}\left(u\right)=1
(47)
\epsilon \left(u,\xi \right)
(49)
\phi \prime \left(u\right)
(50)
\widehat{g}\left(\omega \right)
(51)
{\left|S{C}_{p}\left(u,\xi \right)\right|}^{2}
(53)
\left|\widehat{g}\left(\omega \right)\right|
(55)
{\left|S{C}_{p}\left(u,\xi \right)\right|}^{2}
(56)
$$\xi \left(u\right)=\phi \prime \left(u\right)\text{,}$$
(57)
\phi \prime \left(u\right)
(59)
\phi \prime \left(u\right)
(64)
0.3\text{\hspace{0.17em} mm}
(65)
\lambda =532\text{\hspace{0.17em} nm}
(66)
\text{45 .1 \hspace{0.17em} nm}
(67)
\text{12 \hspace{0.17em} \mu m}\times \text{12 \hspace{0.17em} \mu m}
(68)
\text{1280}\times 256
(69)
3\text{\hspace{0.17em} mm}
(70)
\text{900}\times \text{256}
(72)
t=0.0345\text{\hspace{0.17em}s}
(73)
U\left(x,y;\text{\hspace{0.17em}}{t}_{n}\right)U*\left(x,y;\text{\hspace{0.17em}}{t}_{1}\right)
(75)
90.2\text{\hspace{0.17em}}\text{\mu m}/\text{s}
(76)
200\text{\hspace{0.17em}}\text{\mu m}/\text{s}
(77)
t=0.0345\text{\hspace{0.17em}s}
(81)
f\left(t\right)g\left(t-u\right)
(83)
\pm 283\text{\hspace{0.17em}}\text{\mu m}/\text{s}
(84)
t=0.0345\text{\hspace{0.17em}s}
(85)
t=0.0345\text{\hspace{0.17em}s}
(86)
t=0.0345\text{\hspace{0.17em}s}
(87)
t=0.0345\text{\hspace{0.17em}s}