## Abstract

We observe a non-complementary dark-space produced when two Ronchi-grams, at zero-phase and π -phase, are overlapped and use these dark spaces to quantify Ronchi-grams. Diffraction and multiple beam interference effects narrow the Ronchi fringes created with a coherent point source illumination and prevent accurate determination of the geometrical fringe edges. The dark spaces created when the intensity of two Ronchi grams is added allow assessing the geometrical edge at the dark space middle providing a way to reduce measurement errors. We re-construct the wavefront deformation in a test beam with a 35-term Zernike polynomial. Experimental results are presented.

©2009 Optical Society of America

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### Equations (9)

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(1)
$$\mathrm{TRA}=\frac{2m\pm 1}{4}p$$
(2)
$$\epsilon {\left(i\right)}_{x}=-2F/\#\sum _{k=1}^{35}{a}_{k}\frac{\partial Z{\left(i\right)}_{k}}{\partial x}$$
(2)
$$\epsilon {\left(i\right)}_{y}=-2F/\#\sum _{k=1}^{35}{a}_{k}\frac{\partial Z{\left(i\right)}_{k}}{\partial y}$$
(3)
$$\chi =\sum _{i=1}^{{N}_{x}}{\left(\epsilon {\left(i\right)}_{x,\mathrm{exp}}-\epsilon {\left(i\right)}_{x,\mathrm{thry}}\right)}^{2}\text{}+\sum _{i=1}^{{N}_{y}}{\left(\epsilon {\left(i\right)}_{y,\mathrm{exp}}-\epsilon {\left(i\right)}_{y,\mathrm{thry}}\right)}^{2}$$
(4)
$${A\mathrm{ZZ}}^{T}=-\frac{1}{2F/\#}E{Z}^{T}$$
(5)
$$A=({a}_{1}{a}_{2}\cdots {a}_{35}),$$
(6)
$$E=({\epsilon}_{x}\left(1\right){\phantom{\rule{.2em}{0ex}}\epsilon}_{x}\left(2\right)\phantom{\rule{.2em}{0ex}}\cdots {\phantom{\rule{.2em}{0ex}}\epsilon}_{x}\left({N}_{x}\right){\phantom{\rule{.2em}{0ex}}\epsilon}_{y}\left(1\right)\phantom{\rule{.2em}{0ex}}{\epsilon}_{y}\left(2\right)\phantom{\rule{.2em}{0ex}}\dots {\phantom{\rule{.2em}{0ex}}\epsilon}_{y}\left({N}_{y}\right)),$$
(7)
$$Z=\left(\begin{array}{cccccccc}\frac{\partial Z{\left(1\right)}_{1}}{\partial x}& \frac{\partial Z{\left(2\right)}_{1}}{\partial x}& \cdots & \frac{\partial Z{\left({N}_{x}\right)}_{1}}{\partial x}& \frac{\partial Z{\left(1\right)}_{1}}{\partial y}& \frac{\partial Z{\left(2\right)}_{1}}{\partial y}& \cdots & \frac{\partial Z{\left({N}_{y}\right)}_{1}}{\partial y}\\ \frac{\partial Z{\left(1\right)}_{2}}{\partial x}& \frac{\partial Z{\left(2\right)}_{2}}{\partial x}& \cdots & \frac{\partial Z{\left({N}_{x}\right)}_{2}}{\partial x}& \frac{\partial Z{\left(1\right)}_{2}}{\partial y}& \frac{\partial Z{\left(2\right)}_{2}}{\partial y}& \cdots & \frac{\partial Z{\left({N}_{y}\right)}_{2}}{\partial y}\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \frac{\partial Z{\left(1\right)}_{34}}{\partial x}& \frac{\partial Z{\left(2\right)}_{34}}{\partial x}& \cdots & \frac{\partial Z{\left({N}_{x}\right)}_{34}}{\partial x}& \frac{\partial Z{\left(1\right)}_{34}}{\partial y}& \frac{\partial Z{\left(2\right)}_{34}}{\partial y}& \cdots & \frac{\partial Z{\left({N}_{y}\right)}_{34}}{\partial x}\\ \frac{\partial Z{\left(1\right)}_{35}}{\partial x}& \frac{\partial Z{\left(2\right)}_{35}}{\partial x}& \cdots & \frac{\partial Z{\left({N}_{x}\right)}_{35}}{\partial x}& \frac{\partial Z{\left(1\right)}_{35}}{\partial y}& \frac{\partial Z{\left(2\right)}_{35}}{\partial y}& \cdots & \frac{\partial Z{\left({N}_{y}\right)}_{35}}{\partial y}\end{array}\right)$$
(8)
$$A=-\frac{1}{2F/\#}E{Z}^{T}{\left(Z{Z}^{T}\right)}^{-1}$$