## Abstract

We study the effects of an extended light source on the calibration of an interference microscope, also referred to as an optical profiler. Theoretical and experimental numerical aperture (NA) factors for circular and linear light sources along with collimated laser illumination demonstrate that the shape of the light source or effective aperture cone is critical for a correct NA factor calculation. In practice, more-accurate results for the NA factor are obtained when a linear approximation to the filament light source shape is used in a geometric model. We show that previously measured and derived NA factors show some discrepancies because a circular rather than linear approximation to the filament source was used in the modeling.

© 2004 Optical Society of America

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### Equations (8)

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(1)
$$\overline{m}=\iint \frac{1}{C}\left(pcos\mathrm{\phi}+hsin\mathrm{\phi}cos\mathrm{\chi}\right)\frac{\mathrm{d}\mathrm{\Omega}}{\mathrm{\Omega}},$$
(2)
$$\overline{m}=\frac{p}{C}\frac{1+cosu}{2},$$
(3)
$$U\left(h\right)={\int}_{0}^{u}{P}_{1}\left(\mathrm{\phi}\right){P}_{2}\left(\mathrm{\phi}\right)R\left(\mathrm{\phi}\right)exp\left(-2\mathit{ikd}cos\mathrm{\phi}\right)\times sin\mathrm{\phi}\mathrm{d}\mathrm{\phi},$$
(4)
$$P\left(\mathrm{\phi}\right)={cos}^{n}\mathrm{\phi}.$$
(5)
$$\mathrm{OPD}={\mathit{AT}}_{i}+{T}_{i}P=d/cos\mathrm{\alpha}+dcos\mathrm{\beta}/cos\mathrm{\alpha},$$
(6)
$$\mathrm{\alpha}=2\mathrm{\theta}-\mathrm{\phi},\mathrm{\beta}=\mathrm{\alpha}-\mathrm{\phi}.$$
(7)
$$\overline{m}=\frac{{\displaystyle \sum _{i=1}^{n}}{\mathrm{OPD}}_{i}}{n}.$$
(8)
$$\mathrm{\beta}=1+0.25{\left(\mathrm{NA}\right)}^{2}+0.1{\left(\mathrm{NA}\right)}^{4}+0.086{\left(\mathrm{NA}\right)}^{6}.$$