## Abstract

A wavelength-scanning heterodyne interference confocal microscope quickly accomplishes the simultaneous measurement of the thickness and the refractive index of a sample by detection of the amplitude and the phase of the interference signal during a sample scan. However, the measurement range of the optical path difference (OPD) that is obtained from the phase changes is limited by the time response of the phase-locked loop circuit in the FM demodulator. To overcome this limitation and to improve the accuracy of the separation measurement, we propose an OPD detection using digital signal processing with a Hilbert transform. The measurement range is extended approximately five times, and the resolution of the OPD is improved to 5.5 from 9 µm without the electrical noise of the FM demodulator circuit. By applying this method for simultaneous measurement of thickness and the refractive index, we can measure samples 20–30-µm thick with refractive indices between 1 and 1.5.

© 2002 Optical Society of America

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### Equations (7)

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(1)
$$\mathit{\nu}\left(t\right)={\mathit{\nu}}_{0}+\mathrm{\Delta}\mathit{\nu}cos\left(2\mathrm{\pi}{f}_{m}t+\mathrm{\varphi}\right),$$
(2)
$$f\left(t\right)\approx cos\left[2\mathrm{\pi}\mathit{ft}+\frac{2\mathrm{\pi}}{c}\mathrm{\Delta}\mathit{\nu}\mathrm{\Delta}lcos\left(2\mathrm{\pi}{f}_{m}t+\mathrm{\varphi}\right)+b\right],$$
(3)
$$f={f}_{s}-{f}_{r},b=\frac{2\mathrm{\pi}}{c}\left({\mathit{\nu}}_{0}+{f}_{r}\right){l}_{r}+\frac{2\mathrm{\pi}}{c}\left({\mathit{\nu}}_{0}+{f}_{s}\right){l}_{s},$$
(4)
$$g\left(t\right)=\frac{1}{\mathrm{\pi}}{\int}_{-\infty}^{\infty}\frac{f\left(\mathrm{\tau}\right)}{t-\mathrm{\tau}}\mathrm{d}\mathrm{\tau}$$
(5)
$$h\left(t\right)=2\mathrm{\pi}f+2\mathrm{\pi}{f}_{m}\left(2\mathrm{\pi}/c\right)\mathrm{\Delta}\mathit{\nu}\mathrm{\Delta}lsin\left(2\mathrm{\pi}{f}_{m}t+\mathrm{\varphi}\right).$$
(6)
$$n={\left(\frac{1}{2}\left\{{\mathrm{NA}}^{2}+{\left[{\mathrm{NA}}^{4}+4\left(1-{\mathrm{NA}}^{2}\right){\left(\mathrm{\Delta}l/\mathrm{\Delta}z\right)}^{2}\right]}^{1/2}\right\}\right)}^{1/2},$$
(7)
$$\mathrm{\Delta}n=\left(|{n}_{m}-{n}_{r}|\right)/{n}_{r},$$