## Abstract

The simultaneous quantitative measurement of out-of-plane displacement and slope using the fast Fourier transform method with a single three-aperture digital speckle pattern interferometry (DSPI) arrangement is demonstrated. The method coherently combines two sheared object waves with a smooth reference wave at the CCD placed at the image plane of an imaging lens with a three-aperture mask placed in front of it. The apertures also introduce multiple spatial carrier fringes within the speckle. A fast Fourier transform of the image generates seven distinct diffraction halos in the spectrum. By selecting the appropriate halos, one can directly obtain two independent out-of-plane displacement phase maps and a slope phase map from the two speckle images, one before and the second after loading the object. It is also demonstrated that by subtracting the out-of-plane displacement phase maps one can generate the same slope phase map.
Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center.

© 2007 Optical Society of America

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### Equations (16)

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(1)
$${u}_{0}\left(x,y\right)=\left|{u}_{0}\left(x,y\right)\right|\mathrm{exp}\left[i\left\{2\pi {f}_{0x}x\right\}\right],$$
(2)
$${u}_{1}\left(x,y\right)=\left|{u}_{1}\left(x,y\right)\right|\mathrm{exp}\left[i\left\{\varphi \left(x,y\right)\right\}\right],$$
(3)
$${u}_{2}\left(x,y\right)=\left|{u}_{1}\left(x+\Delta x,y\right)\right|\mathrm{exp}\left[i\left\{\varphi \left(x+\Delta x,y\right)+2\pi {f}_{0y}y\right\}\right]\text{,}$$
(4)
$$I\left(x,y\right)={\left|{u}_{0}\left(x,y\right)+{u}_{1}\left(x,y\right)+{u}_{2}\left(x,y\right)\right|}^{2}={u}_{0}{u}_{0}*+{u}_{1}{u}_{1}*+{u}_{2}{u}_{2}*+{u}_{0}{u}_{1}*+{u}_{1}{u}_{0}*+{u}_{0}{u}_{2}*+{u}_{2}{u}_{0}*+{u}_{1}{u}_{2}*+{u}_{2}{u}_{1}*.$$
(5)
$$I\left(x,y\right)={\left|{u}_{0}\left(x,y\right)\right|}^{2}+{\left|{u}_{1}\left(x,y\right)\right|}^{2}+{\left|{u}_{1}\left(x+\Delta x,y\right)\right|}^{2}+\left|{u}_{0}\left(x,y\right)\right|\cdot \left|{u}_{1}\left(x,y\right)\right|\mathrm{exp}\left[i\left\{2\pi {f}_{0}x-\varphi \left(x,y\right)\right\}\right]+\left|{u}_{0}\left(x,y\right)\right|\cdot \left|{u}_{1}\left(x,y\right)\right|\mathrm{exp}[-i\left\{2\pi {f}_{0}x-\varphi \left(x,y\right)\right\}]+\left|{u}_{0}\left(x,y\right)\right|\cdot \left|{u}_{1}\left(x+\Delta x,y\right)\right|\mathrm{exp}\left[i\left\{2\pi {f}_{0}x-2\pi {f}_{0}y-\varphi \left(x+\Delta x,y\right)\right\}\right]+\left|{u}_{0}\left(x,y\right)\right|\cdot |{u}_{1}\left(x+\Delta x,y\right)\times |\mathrm{exp}\left[-i\left\{2\pi {f}_{0}x-2\pi {f}_{0}y-\varphi (x+\Delta x,y)\right\}\right]+\left|{u}_{1}\left(x,y\right)\right|\cdot \left|{u}_{1}\left(x+\Delta x,y\right)\right|\mathrm{exp}\left[i\left\{\varphi \left(x,y\right)-\varphi \left(x+\Delta x,y\right)-2\pi {f}_{0}y\right\}\right]+\left|{u}_{1}\left(x,y\right)\right|\cdot |{u}_{1}\left(x+\Delta x,y\right)\times |\mathrm{exp}\left[-i\left\{\varphi \left(x,y\right)-\varphi \left(x+\Delta x,y\right)-2\pi {f}_{0}y\right\}\right]\text{.}$$
(6)
$$i=FT\left[I\right]={U}_{0}\otimes {U}_{0}*+{U}_{1}\otimes {U}_{1}*+{U}_{2}\otimes {U}_{2}*+{U}_{0}\otimes {U}_{1}*+{U}_{1}\otimes {U}_{0}*+{U}_{0}\otimes {U}_{2}*+{U}_{2}\otimes {U}_{0}*+{U}_{1}\otimes {U}_{2}*+{U}_{2}\otimes {U}_{1}*\text{,}$$
(7)
$$\varphi \left(x,y\right)-2\pi {f}_{0}x=\mathrm{arctan}\left[\frac{\mathrm{Im}\left({u}_{1}{u}_{0}*\right)}{\mathrm{Re}\left({u}_{1}{u}_{0}*\right)}\right]=\mathrm{arctan}\left[\frac{{N}_{B}}{{D}_{B}}\right]\text{,}$$
(8)
$$\varphi \left(x,y\right)-\varphi \left(x+\Delta x,y\right)-2\pi {f}_{0}y=\mathrm{arctan}\left[\frac{\mathrm{Im}\left({u}_{1}{u}_{2}*\right)}{\mathrm{Re}\left({u}_{1}{u}_{2}*\right)}\right]$$
(9)
$$=\mathrm{arctan}\left[\frac{{N}_{B}\prime}{{D}_{B}\prime}\right]\text{,}$$
(10)
$${u}_{1}\left(x,y\right)=\left|{u}_{1}\left(x,y\right)\right|\mathrm{exp}\left[i\left\{\varphi \prime \left(x,y\right)\right\}\right],$$
(11)
$${u}_{2}\left(x,y\right)=\left|{u}_{1}\left(x+\Delta x,y\right)\right|\mathrm{exp}\left[i\left\{\varphi \prime \left(x+\Delta x,y\right)+2\pi {f}_{0}y\right\}\right]\text{.}$$
(12)
$$\Delta \varphi \left(x,y\right)=\varphi \prime \left(x,y\right)-\varphi \left(x,y\right)=\left[\varphi \prime \left(x,y\right)-2\pi {f}_{0}x\right]-\left[\varphi \left(x,y\right)-2\pi {f}_{0}x\right]=\mathrm{arctan}\left(\frac{{N}_{A}{D}_{B}-{N}_{B}{D}_{A}}{{N}_{A}{N}_{B}+{D}_{A}{D}_{B}}\right)=\mathrm{arctan}\left(\frac{N}{D}\right)\text{.}$$
(13)
$$\Delta \varphi \left(x,y\right)=\frac{4\pi}{\lambda}\text{\hspace{0.17em}}w\left(x,y\right).$$
(14)
$$\Delta \varphi \prime \left(x,y\right)=\left[\varphi \prime \left(x,y\right)-\varphi \left(x,y\right)\right]-\left[\varphi \prime (x+\mathrm{\Delta}x,y)-\varphi \left(x+\Delta x,y\right)\right]=\left[\varphi \prime \left(x,y\right)-\varphi \prime \left(x+\Delta x,y\right)-2\pi {f}_{0}y\right]-\left[\varphi \left(x,y\right)-\varphi \left(x+\Delta x,y\right)-2\pi {f}_{0}y\right]=\mathrm{arctan}\left(\frac{{N}_{A}\prime {D}_{B}\prime -{N}_{B}\prime {D}_{A}\prime}{{N}_{A}\prime {N}_{B}\prime +{D}_{A}\prime {D}_{B}\prime}\right)=\mathrm{arctan}\left(\frac{N\prime}{D\prime}\right)\text{,}$$
(15)
$$\Delta \varphi \prime \left(x,y\right)=\frac{4\pi}{\lambda}\text{\hspace{0.17em}}\frac{\partial w\left(x,y\right)}{\partial x}\text{\hspace{0.17em}}\Delta x,$$
(16)
$$\Delta \varphi \text{\u2033}\left(x,y\right)=\Delta \varphi \left(x+\Delta x,y\right)-\Delta \varphi \left(x,y\right)=\frac{4\pi}{\lambda}\left[\frac{w\left(x+\Delta x,y\right)-w\left(x,y\right)}{\Delta x}\right]\Delta x=\frac{4\pi}{\lambda}\text{\hspace{0.17em}}\frac{\partial w\left(x,y\right)}{\partial x}\text{\hspace{0.17em}}\Delta x.$$