## Abstract

The confocal-detection principle is open especially for use in medical applications. For inspection systems applications for technical objects in reflection confocal setups are of growing importance. For such applications the confocal measurements need to have a very short measuring time. A fast detection system is needed and to satisfy this requirement only a small number of height levels are measured and a fast-evaluation algorithm is used. Drawbacks of the reduction of height levels are a greater influence of noise and additional systematic errors on the measured heights. Study the effects of the reduction are calculated, different evaluation algorithms are analyzed, and the optimization of the parameters is discussed.

© 2002 Optical Society of America

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### Equations (10)

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(1)
$$I\left(z\right)\propto {\mathrm{sinc}}^{2}\left(\mathit{az}\right),$$
(2)
$$C=\frac{\sum \mathit{zI}\left(z\right)}{\sum I\left(z\right)}.$$
(3)
$$\mathrm{\sigma}_{C}{}^{2}={k}^{2}\left(z\right)\mathrm{\sigma}_{z}{}^{2}+{m}^{2}\left(I\right)\mathrm{\sigma}_{I}{}^{2},$$
(4)
$$k={\left[\sum _{j}{\left(\frac{\partial C}{\partial {z}_{j}}\right)}^{2}\right]}^{1/2}={\left\{{\left(\frac{1}{\sum _{i}I\left({z}_{i}\right)}\right)}^{2}\times \sum _{j}{\left[I\left({z}_{j}\right)+\left({z}_{j}-C\right)I\prime \left({z}_{j}\right)\right]}^{2}\right\}}^{1/2},$$
(5)
$$m={\left[\sum _{j}{\left(\frac{\partial C}{\partial {I}_{j}}\right)}^{2}\right]}^{1/2}={\left[{\left(\frac{1}{\sum _{i}{I}_{i}}\right)}^{2}\sum _{j}{\left({z}_{j}-C\right)}^{2}\right]}^{1/2},$$
(6)
$$k=\left(\frac{1}{\sum _{i}{\mathrm{sinc}}^{2}\left({z}_{i}\right)-h}\right){\left\{\sum _{j}{\left[2cos\left({z}_{j}\right)\mathrm{sinc}\left({z}_{j}\right)+{\mathrm{sinc}}^{2}\left({z}_{j}\right)-h\right]}^{2}\right\}}^{1/2},$$
(7)
$$m=\left(\frac{1}{{\displaystyle \sum _{i}}{\mathrm{sinc}}^{2}\left({z}_{i}\right)-h}\right){\left(\sum _{j}z_{j}{}^{2}\right)}^{1/2}.$$
(8)
$$\mathrm{\Delta}{z}_{c}=\frac{N}{{\left[{\displaystyle \sum _{j}}I{\prime}^{2}\left({z}_{j}-{z}_{c}\right)\right]}^{1/2}},$$
(9)
$$a=\frac{{x}_{1}}{{x}_{1}+{x}_{2}+{x}_{3}},b=\frac{{x}_{3}}{{x}_{1}+{x}_{2}+{x}_{3}},$$
(10)
$$a=\left\{\begin{array}{cc}\frac{{x}_{2}-{x}_{1}}{{x}_{2}+{x}_{3}-{x}_{1}}& {x}_{3}>{x}_{1}\\ \frac{{x}_{2}-{x}_{3}}{{x}_{2}+{x}_{1}-{x}_{3}}& {x}_{3}\le {x}_{1}\end{array}\right.,b=\frac{{x}_{2}-{x}_{3}}{2{x}_{2}-{x}_{1}-{x}_{3}}.$$