## Abstract

The effects of an arbitrary small inclination between the two
crossed gratings on moiré fringes in Talbot interferometry are
discussed. The small inclination is formed by the rotation by a
small angle γ of the beam splitter’s grating about the axis that is
on the plane of the grating and has an arbitrary angle δ with respect
to the lines of the grating. The results indicate that the small
inclination has a great influence on measurements for which Talbot
interferometry is applied, such as beam collimation and measurement of
the focal length of a lens. The theoretical analyses are proved by
experimental results. Some methods for judging the size of a small
inclination are also proposed.

© 2001 Optical Society of America

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### Equations (9)

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(1)
$$T\left({x}_{1},{y}_{1}\right)=\left[{A}_{0}+2{A}_{1}cos\left(2\mathrm{\pi}{x}_{1}/d\right)\right]\times \left[{B}_{0}+{B}_{1}cos\left(2\mathrm{\pi}{y}_{1}/d\right)\right],$$
(2)
$$T\left({x}_{0},{y}_{0}\right)=\left[{A}_{0}+2{A}_{1}cos\left(2\mathrm{\pi}{x}_{0}/{d}_{x0}\right)\right]\times \left[{B}_{0}+{B}_{1}cos\left(2\mathrm{\pi}{y}_{0}/{d}_{y0}\right)\right],$$
(3)
$${d}_{x0}={\left[{\left({d}_{R\perp}cos\mathrm{\gamma}\right)}^{2}+d_{R\Vert}{}^{2}\right]}^{1/2}=d{\left({sin}^{2}\mathrm{\delta}{cos}^{2}\mathrm{\gamma}+{sin}^{2}\mathrm{\delta}\right)}^{1/2},$$
(4)
$${d}_{y0}={\left[{\left({d}_{R\perp}cos\mathrm{\gamma}\right)}^{2}+d_{R\Vert}{}^{2}\right]}^{1/2}=d{\left({cos}^{2}\mathrm{\delta}{cos}^{2}\mathrm{\gamma}+{sin}^{2}\mathrm{\delta}\right)}^{1/2},$$
(5)
$$I\left(x,y,{z}_{T}\right)=I\left(x,{z}_{T}\right)I\left(y,{z}_{T}\right)={\left[{A}_{0}+2{A}_{1}cos\left(2\mathrm{\pi}x/{d}_{x0}\right)\right]}^{2}\times {\left[{B}_{0}+{B}_{1}cos\left(2\mathrm{\pi}y/{d}_{y0}\right)\right]}^{2}.$$
(6)
$$I\prime ={I}_{x}\prime {I}_{y}\prime =\left[{A}_{0}+2{A}_{1}cos\left(2\mathrm{\pi}/d\right)\left(xcos\mathrm{\theta}-ysin\mathrm{\theta}\right)\right]\left[{B}_{0}+2{B}_{1}cos\left(2\mathrm{\pi}/d\right)\left(xsin\mathrm{\theta}-ycos\mathrm{\theta}\right)\right].$$
(7)
$${I}_{\mathit{xM}}\left(x,y\right)\propto A_{0}{}^{3}+4{A}_{0}A_{1}{}^{2}cos2\mathrm{\pi}\left[\left(\frac{cos\mathrm{\theta}}{d}-\frac{1}{{d}_{0}}\right)x[-\frac{1}{d}ysin\mathrm{\theta}\right].$$
(8)
$$tan{\mathrm{\varphi}}_{x}=cot\mathrm{\theta}-\frac{1}{sin\mathrm{\theta}{\left({sin}^{2}\mathrm{\delta}{cos}^{2}\mathrm{\gamma}+{cos}^{2}\mathrm{\delta}\right)}^{1/2}}.$$
(9)
$$tan{\mathrm{\varphi}}_{y}=cot\mathrm{\theta}-\frac{1}{sin\mathrm{\theta}{\left({cos}^{2}\mathrm{\gamma}{cos}^{2}\mathrm{\delta}+{sin}^{2}\mathrm{\delta}\right)}^{1/2}},$$